Finite Group Databasebeta

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#72318572

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#139828
commutatorC23×D4C22×C4
Frattini, cosocleC2×#535C23
prime radical#35527C22
LCSG ▷ C23×D4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #35527 ◁ G
LECSG ▷ C2×#535 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

1536 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 1536 conjugacy classes of subgroups — too many to draw. Only its 77 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:

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

Polycyclic presentation:

2^10|b^2=f,c^2=gj,h^2=j,b^a=bj,c^a=cd,c^b=ce,d^b=dg,e^a=eg,e^b=eghj,e^c=ej,f^c=fgh,f^d=fj,f^e=fi,g^b=gj,h^a=hj,h^b=hi,h^c=hj,h^e=hj,h^f=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

49 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72318585, #72318830, #72318837, …
maximal normal subgroup of#72318585, #72318830, #72318837, …

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