Finite Group Databasebeta

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#72317999

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#148170
commutator#532C22×C4
Frattini, cosocleC2×#532C23
prime radicalC2×#3549C22
LCSG ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G
LECSG ▷ C2×#532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

2076 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 2076 conjugacy classes of subgroups — too many to draw. Only its 139 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)(3,12,15,25)(4,17)(5,10,21,8)(6,11)(7,22,23,14)(9,20,28,32)(13,29,16,30)(18,27)(19,24,31,26);(4,18)(5,21)(7,23)(9,28)(12,29)(14,31)(17,27)(19,22)(24,26)(25,30);(1,3,6,16)(2,7)(4,19,18,22)(5,13,21,15)(8,23)(9,12)(10,24)(11,26)(14,32,31,20)(17,25)(27,29)(28,30)

Polycyclic presentation:

2^10|a^2=g,c^2=egi,f^2=i,h^2=j,b^a=be,c^a=cefi,c^b=cd,d^a=dh,e^c=eh,e^d=eij,f^a=fh,f^b=fj,f^c=fh,f^d=fij,f^e=fij,g^c=ghj,g^d=gij,g^f=gij,h^a=hi,h^b=hi,h^c=hj,h^d=hj,h^e=hj,h^g=hj,i^a=ij,i^c=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72318015, #72318062, #72318165, …
maximal normal subgroup of#72318015, #72318062, #72318165, …

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