Finite Group Databasebeta

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#72238437

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#8534
commutatorC23×D4C22×C4
Frattini, cosocleC22×#120C23
prime radical#4465748C2
LCSG ▷ C23×D4 ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#120 ◁ G
LECSG ▷ C22×#120 ▷ C24 ▷ C22 ▷ C1

2594 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 2594 conjugacy classes of subgroups — too many to draw. Only its 91 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:

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

Polycyclic presentation:

2^10|a^2=d,b^2=gij,c^2=def,d^2=ij,b^a=bde,c^a=cef,c^b=cdegij,d^b=dg,d^c=dh,e^a=egj,e^b=egij,e^c=eg,e^d=ei,f^a=fghj,f^b=fgh,f^c=fgh,f^d=fij,g^a=gi,g^c=gi,h^a=hj,h^c=hj

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72238440
maximal normal subgroup of#72238440

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