Finite Group Databasebeta

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#72324306

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#8535
commutatorC23×D4C22×C4
Frattini, cosocleC22×#121C23
prime radical#5486275C2
LCSG ▷ C23×D4 ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#121 ◁ G
LECSG ▷ C22×#121 ▷ C24 ▷ C22 ▷ C1

2176 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 2176 conjugacy classes of subgroups — too many to draw. Only its 85 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=h,b^a=bd,c^a=ce,c^b=cg,d^a=dh,d^b=dj,d^c=dg,e^a=ehij,e^b=eg,e^d=ei,f^b=fh,f^c=fhij,f^d=fj,f^e=fj,g^a=gi,h^a=hj

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72324308, #72324495, #72324498
maximal normal subgroup of#72324308, #72324495, #72324498

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