Finite Group Databasebeta

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#72321467

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#130798
commutatorC22×C16C22×C4
Frattini, cosocleC2×#504C23
prime radical#7623852C2
LCSG ▷ C22×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ #33435 ◁ G
LECSG ▷ C2×#504 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

790 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 790 conjugacy classes of subgroups — too many to draw. Only its 111 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=g,c^2=dfi,d^2=hi,e^2=hij,f^2=ij,h^2=ij,i^2=j,b^a=bg,c^a=ce,c^b=cd,d^a=dh,d^b=dhj,d^c=dg,e^a=eh,e^b=ehj,f^a=fi,f^b=fgi,f^c=fg,f^d=fj,g^c=gj,h^a=hi,h^b=hi,i^a=ij,i^b=ij

Conjugacy classes & character table

67 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72321495, #72342917, #72342966
maximal normal subgroup of#72321495, #72342917, #72342966

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