Finite Group Databasebeta

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#72342808

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Subgroups

subgroupG/H
centerC4#10336
commutatorC23×D4C22×C4
Frattini, cosocleC23×#41C23
socleC2#116351
prime radical#62811C22
LCSG ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C4 ◁ C23×#41 ◁ G
LECSG ▷ C23×#41 ▷ C23 ▷ C2 ▷ C1

2949 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 2949 conjugacy classes of subgroups — too many to draw. Only its 155 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:

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

Polycyclic presentation:

2^10|a^2=d,b^2=fg,c^2=j,d^2=j,e^2=j,b^a=bf,c^a=cgi,c^b=ce,d^b=dh,d^c=dj,e^a=ei,e^b=ehi,e^c=ej,e^d=ej,f^a=fh,f^b=fh,f^c=fhi,f^e=fj,g^b=gh,g^c=gj,g^e=gj,h^c=hj,i^a=ij,i^c=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72342816, #72342848, #72342870
maximal normal subgroup of#72342816, #72342848, #72342870

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