Finite Group Databasebeta

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#72258366

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C25×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#116351
commutatorC23×D4C22×C4
Frattini, cosocleC23×#41C23
prime radicalC22×#535C22
LCSG ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22×C4 ◁ C23×#41 ◁ G
LECSG ▷ C23×#41 ▷ C23 ▷ C2 ▷ C1

3293 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 3293 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:

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

Polycyclic presentation:

2^10|b^2=f,c^2=g,d^2=j,b^a=bd,c^a=cdfg,c^b=ce,d^a=dj,d^b=dj,d^c=dh,e^a=ei,e^b=ehj,e^c=ehij,e^d=ej,f^c=fhj,f^e=fj,g^b=ghij,g^e=gj,h^b=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72258377, #72258400, #72258427
maximal normal subgroup of#72258377, #72258400, #72258427

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