Finite Group Databasebeta

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#72319664

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#149280
commutatorC2×#127C22×C4
Frattini, prime radical, cosocleC22×#127C23
LCSG ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C24 ◁ C23×C4 ◁ C22×#127 ◁ G
LECSG ▷ C22×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

1322 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 1322 conjugacy classes of subgroups — too many to draw. Only its 147 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)(3,7)(4,15,17,8)(5,20)(6,9)(10,21)(11,28,16,32)(12,31)(13,23)(14,24)(18,27,19,26)(22,29,25,30);(1,3)(2,7)(4,16,19,30)(5,13,21,12)(6,14)(8,25,27,28)(9,24)(10,23,20,31)(11,18,29,17)(15,22,26,32);(1,4)(2,8)(3,11,14,16)(5,19)(6,17)(7,22,24,25)(9,15)(10,27)(12,29,13,30)(18,21)(20,26)(23,28,31,32)

Polycyclic presentation:

2^10|a^2=e,b^2=g,c^2=eij,h^2=j,b^a=bg,c^a=cd,c^b=cf,d^a=dj,d^b=dh,e^c=ej,f^a=fi,f^b=fhi,f^c=fj,f^d=fj,g^c=ghij,g^d=gij,g^f=gi,h^a=hi,h^b=hi,h^d=hj,h^f=hj,h^g=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72319671, #72319945
maximal normal subgroup of#72319671, #72319945

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