Finite Group Databasebeta

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#72306559

order1024
nilpotency class6
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C25×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2C2×#13673
commutator, Frattini, cosocle#493C24
LCSG ▷ #493 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #3147 ◁ G
LECSG ▷ #493 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

1487 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 1487 conjugacy classes of subgroups — too many to draw. Only its 115 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)(4,18)(5,11)(6,13)(7,14)(8,12)(10,22)(15,29)(16,26)(17,27)(19,28)(20,25)(21,31)(23,32)(24,30);(1,3)(2,9)(4,19,24,20,22,23,21,15)(5,7)(6,16)(8,17)(10,28,31,25,18,32,30,29)(11,14)(12,27)(13,26);(4,20,21,19,22,15,24,23)(10,29,30,32,18,25,31,28);(1,4)(2,10)(3,15)(5,23)(6,21)(7,24)(8,22)(9,25)(11,28)(12,18)(13,30)(14,31)(16,19)(17,20)(26,32)(27,29)

Polycyclic presentation:

2^10|b^2=ghij,c^2=gh,e^2=h,g^2=i,h^2=j,b^a=bij,d^a=dj,d^b=df,d^c=de,e^b=eg,e^d=ehj,f^a=fj,f^b=fhj,f^c=fgh,f^e=fh,g^b=gi,g^d=gh,g^f=gi,h^b=hi,h^d=hj,h^f=hj,i^d=ij

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72306579, #72338309, #72338390
maximal normal subgroup of#72306579, #72338309, #72338390

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