Finite Group Databasebeta

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#72321288

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#130718
commutatorC22×C16C22×C4
Frattini, cosocleC2×#504C23
prime radical#7623644C2
LCSG ▷ C22×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ #33434 ◁ G
LECSG ▷ C2×#504 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

1178 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 1178 conjugacy classes of subgroups — too many to draw. Only its 115 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) = 64
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(2,8)(3,14)(4,18)(6,20)(7,24)(9,28)(10,27)(11,35)(12,37)(13,30)(15,41)(16,46)(19,25)(21,48)(22,54)(23,53)(26,29)(31,60)(32,58)(33,61)(34,59)(36,55)(38,44)(39,43)(42,57)(45,64)(47,51)(49,50)(52,56)(62,63);(1,2)(3,10,14,9)(4,19)(5,13)(6,25)(7,26)(8,18)(11,20)(12,24)(15,42,38,51)(16,47,39,57)(17,32,40,34)(21,52)(22,55)(23,56)(27,44,28,41)(29,54)(30,48)(31,43,33,46)(35,49)(36,53)(37,50)(45,61,62,60)(58,63,59,64);(2,9)(3,14)(8,27)(10,13)(11,31)(12,32)(15,38)(16,39)(17,40)(19,51)(25,57)(26,34)(28,30)(29,58)(33,36)(35,61)(37,59)(41,44)(42,52)(43,46)(45,62)(47,56)(55,60)(63,64)

Polycyclic presentation:

2^10|b^2=f,d^2=j,e^2=hi,h^2=ij,i^2=j,b^a=be,c^a=cgj,c^b=cd,d^a=dj,d^b=dg,d^c=dj,e^a=eh,e^b=eh,f^c=fgj,f^d=fj,g^b=gj,h^a=hi,h^b=hi,i^a=ij,i^b=ij

Conjugacy classes & character table

73 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72321336, #72342920, #72343025
maximal normal subgroup of#72321336, #72342920, #72343025

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, 2, F.6);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, 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.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);