Finite Group Databasebeta

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#72322766

order1024
nilpotency class6
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#153018
commutatorC2×C32C22×C4
Frattini, cosocleC22×C32C23
prime radicalC23#2263
LCSG ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ #13549 ◁ G
LECSG ▷ C22×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

267 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 267 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 128
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 3 generators on 128 points:

(1,2,6,13,33,61,60,104,9,15,41,69,49,73,98,107,45,51,37,65,100,101,125,32,75,4,25,28,8,14,40,68,7,17,5,12,34,62,67,109,44,53,36,64,72,117,96,113,10,16,42,70,105,29,93,39,48,30,92,27,43,47,35,63)(3,18,20,77,85,95,76,57,22,79,86,99,89,123,103,74,52,119,120,83,116,84,106,82,114,110,111,56,21,59,31,55,24,78,88,71,46,97,91,26,54,94,121,127,115,124,108,58,23,80,87,128,118,126,112,19,50,81,102,122,90,66,38,11);(1,3)(2,11,17,55)(4,26,30,57)(5,31)(6,38)(7,24)(8,46)(9,50)(10,52)(12,59,13,66)(14,71,47,77)(15,19,53,82)(16,74,51,58)(18,68,78,63)(20,35)(21,34)(22,48)(23,45)(25,91)(27,95,28,97)(29,99,101,127)(32,94,39,79)(33,90)(36,106)(37,108)(40,88)(41,112)(42,103)(43,85)(44,114)(49,118)(54,75)(56,61,122,62)(60,102)(64,84,69,126)(65,124,70,123)(67,111)(72,116)(73,128,117,83)(76,92)(80,113,119,107)(81,109,110,104)(86,93)(87,98)(89,105)(96,120)(100,115)(121,125);(1,4,10,53,7,30,45,15)(2,9,51,48,17,44,16,75)(3,19,23,57,24,82,52,26)(5,32,37,109,6,39,42,104)(8,47,49,73,43,14,72,117)(11,54,74,114,55,22,58,50)(12,60,70,93,13,67,65,125)(18,76,119,112,78,91,80,106)(20,84,87,97,88,126,120,95)(21,56,89,127,90,122,115,99)(25,68,41,113,92,63,36,107)(27,96,69,40,28,98,64,35)(29,100,61,33,101,105,62,34)(31,81,103,79,38,110,108,94)(46,83,116,71,85,128,118,77)(59,121,124,111,66,86,123,102)

Polycyclic presentation:

2^10|a^2=e,b^2=defgj,c^2=defgij,d^2=ghj,e^2=ghj,g^2=hi,h^2=ij,i^2=j,b^a=bd,c^a=cdi,c^b=cfij,d^b=dg,d^c=dg,e^b=egj,e^c=eg,f^a=fj,f^b=fj,f^c=fj,g^b=gh,g^c=gh,h^b=hi,h^c=hi,i^b=ij,i^c=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72322773, #72322907, #72322982
maximal normal subgroup of#72322773, #72322907, #72322982

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.4*F.5*F.6*F.7*F.10);;
gap> SetPower(coll, 3, F.4*F.5*F.6*F.7*F.9*F.10);;
gap> SetPower(coll, 4, F.7*F.8*F.10);;
gap> SetPower(coll, 5, F.7*F.8*F.10);;
gap> SetPower(coll, 7, F.8*F.9);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.4*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);