Finite Group Databasebeta

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#72338695

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#251883
commutator, Frattini, cosocleC23×D4C24
LCSG ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ G
LECSG ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1

1628 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 1628 conjugacy classes of subgroups — too many to draw. Only its 127 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:

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

Polycyclic presentation:

2^10|b^2=h,d^2=gi,e^2=j,b^a=bh,c^a=ce,c^b=ceg,d^a=df,d^b=dfj,d^c=dgi,e^a=ej,e^b=ei,e^c=ej,e^d=eh,f^b=fj,f^c=fhi,f^d=fij,g^a=gij,g^b=gij,g^c=gj,g^e=gj,h^c=hj,h^d=hj,i^c=ij

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72338701, #72338728, #72338789
maximal normal subgroup of#72338701, #72338728, #72338789

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