Finite Group Databasebeta

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#72306550

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#246927
commutator, Frattini, cosocleC4×M16C24
prime radical#7487865C2
LCSG ▷ C4×M16 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ G
LECSG ▷ C4×M16 ▷ C2×C4 ▷ C2 ▷ C1

1955 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 1955 conjugacy classes of subgroups — too many to draw. Only its 151 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)(4,16)(7,21)(9,17)(12,27)(13,19)(14,28)(15,29)(18,31)(22,26)(23,30)(25,32);(1,2)(3,12,15,25,11,27,29,32)(4,17)(5,8)(6,10)(7,14,23,18,21,28,30,31)(9,16)(13,26)(19,22)(20,24);(1,3,13,14,6,15,4,18,5,11,19,28,20,29,16,31)(2,7,22,12,10,23,9,25,8,21,26,27,24,30,17,32);(2,8)(4,19)(6,20)(7,21)(9,22)(12,25)(13,16)(14,31)(15,29)(17,26)(18,28)(27,32)

Polycyclic presentation:

2^10|b^2=g,c^2=efghj,e^2=hij,g^2=ij,h^2=j,c^a=cf,c^b=ce,d^b=dgj,d^c=dghij,e^a=ei,e^b=ej,e^c=eh,e^d=ehij,f^b=fij,f^c=fi,f^e=fj,g^c=ghij,g^d=gij,h^d=hj,i^c=ij

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72306554, #72306775
maximal normal subgroup of#72306554, #72306775

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