Finite Group Databasebeta

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#72228064

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22C2×#2196
commutator, Frattini, cosocle#391C24
LCSG ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #2297 ◁ G
LECSG ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

2160 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 2160 conjugacy classes of subgroups — too many to draw. Only its 123 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)(3,9)(4,20)(5,15)(6,28)(7,30)(8,31)(10,19)(11,17)(12,21)(13,25)(14,26)(16,22)(18,23)(24,29)(27,32);(1,3,9,2)(4,16,8,18)(5,24,26,28)(6,14,29,15)(7,13,27,12)(10,17,11,19)(20,23,31,22)(21,32,25,30);(1,4,9,8)(2,12,3,13)(5,21,26,25)(6,27,29,7)(10,22,11,23)(14,16,15,18)(17,28,19,24)(20,32,31,30);(1,5,10,27)(2,13,19,28)(3,12,17,24)(4,6,22,25)(7,9,26,11)(8,29,23,21)(14,16,30,20)(15,18,32,31)

Polycyclic presentation:

2^10|b^2=h,c^2=h,d^2=hj,e^2=h,f^2=j,g^2=j,b^a=bh,c^a=cf,c^b=ce,d^a=dg,d^b=dhj,d^c=dj,e^a=eg,e^b=eh,e^c=eh,f^a=fj,f^b=fgij,f^c=fj,f^d=fij,f^e=fij,g^a=gj,g^c=gi,g^d=gj,g^e=gj,g^f=gj,i^a=ij,i^b=ij

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72228073, #72339097
maximal normal subgroup of#72228073, #72339097

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