Finite Group Databasebeta

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#72313536

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C28

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#245225
commutator, Frattini, cosocleC23×D4C24
LCSG ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ G
LECSG ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1

4304 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 4304 conjugacy classes of subgroups — too many to draw. Only its 157 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=e,d^2=j,b^a=be,c^a=cf,c^b=cfg,d^a=dg,d^b=dj,e^c=eh,e^d=ei,f^a=fh,f^b=fij,g^a=gi,g^b=gij,g^e=gj,h^b=hj,i^a=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72313556, #72314648, #72314655
maximal normal subgroup of#72313556, #72314648, #72314655

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