Finite Group Databasebeta

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#72251815

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#139773
commutatorC2×#127C22×C4
Frattini, cosocleC2×#535C23
prime radical#499832C2
LCSG ▷ C2×#127 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ C2×#535 ◁ G
LECSG ▷ C2×#535 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

1614 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 1614 conjugacy classes of subgroups — too many to draw. Only its 73 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) = 32
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2,8,3)(4,13,19,9)(5,20,21,26)(6,12,7,11)(10,23,14,22)(15,29,24,31)(16,30,28,32)(17,27,18,25);(1,3,8,2)(4,15,17,30)(5,20)(6,11)(7,12)(9,16,25,31)(10,23,14,22)(13,28,27,29)(18,32,19,24)(21,26);(1,4,6,18,8,19,7,17)(2,9,11,25,3,13,12,27)(5,16,23,29,21,28,22,31)(10,24,26,30,14,15,20,32)

Polycyclic presentation:

2^10|a^2=j,b^2=eghj,c^2=ef,d^2=ij,g^2=j,h^2=j,b^a=be,c^a=cgh,c^b=cd,d^a=dg,d^b=dgj,d^c=dghj,e^c=egi,e^d=eij,f^a=fj,f^b=fghi,f^c=fgi,f^d=fi,f^e=fj,g^a=gj,g^b=gi,g^e=gj,g^f=gj,h^a=hj,h^b=hi,h^c=hj,h^e=hj,h^f=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

49 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72251816, #72251821, #72251824, …
maximal normal subgroup of#72251816, #72251821, #72251824, …

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