Finite Group Databasebeta

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#72340173

order1024
nilpotency class6
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#152185
commutator#492C22×C4
Frattini, cosocle#3147C23
prime radical#6369161C2
LCSG ▷ #492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #3147 ◁ G
LECSG ▷ #3147 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

1030 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 1030 conjugacy classes of subgroups — too many to draw. Only its 75 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:

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

Polycyclic presentation:

2^10|a^2=ij,b^2=ef,c^2=gj,d^2=hj,e^2=j,g^2=i,h^2=j,b^a=be,c^a=cgi,c^b=cd,d^a=dg,d^b=dh,d^c=dj,e^c=eg,e^d=eh,f^a=fj,f^c=fg,f^d=fh,g^a=gi,g^b=gh,g^e=gi,g^f=gi,h^a=hi,h^b=hj,h^e=hj,h^f=hj,i^b=ij

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72340175
maximal normal subgroup of#72340175

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