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#72268326

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#23900
commutator, Frattini, cosocleC23×Q8C24
prime radical#62819C22
LCSG ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊Q8 ◁ G
LECSG ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1

2599 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 2599 conjugacy classes of subgroups — too many to draw. Only its 161 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) = 48
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,5,10)(3,6,11,8)(4,7,13,9)(12,16,14,15)(17,18,22,26)(19,23,29,38)(20,24,32,41)(21,33,35,25)(27,36,30,39)(28,42,34,37)(31,43,46,40)(44,48,45,47);(2,6,10,8)(4,13)(7,16,9,15)(12,14)(17,19,22,29)(18,23,26,38)(20,30,32,27)(21,34,35,28)(24,39,41,36)(25,42,33,37)(31,44,46,45)(40,47,43,48);(2,7,10,9)(3,11)(6,15,8,16)(12,14)(17,20,22,32)(18,24,26,41)(19,27,29,30)(21,31,35,46)(23,36,38,39)(25,43,33,40)(28,44,34,45)(37,48,42,47);(17,21)(18,25)(19,28)(20,31)(22,35)(23,37)(24,40)(26,33)(27,44)(29,34)(30,45)(32,46)(36,47)(38,42)(39,48)(41,43)

Polycyclic presentation:

2^10|a^2=ij,b^2=hij,c^2=hij,e^2=j,f^2=j,b^a=be,c^a=cf,c^b=chij,d^a=di,d^b=di,d^c=dg,e^a=ej,e^c=eh,f^a=fj,f^b=fh,f^d=fi,f^e=fj,g^a=gi,h^a=hj

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72268331
maximal normal subgroup of#72268331

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