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#72262803

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C27

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#23984
commutator, Frattini, cosocleC22×C22⋊C4C24
LCSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#120 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

3329 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 3329 conjugacy classes of subgroups — too many to draw. Only its 137 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)(3,7)(4,15)(5,14)(6,12)(8,10)(9,17)(11,23)(13,20)(16,28)(18,31)(19,22)(21,27)(24,30)(25,32)(26,29);(1,3,6,10)(2,8,12,7)(4,9,18,26)(5,19,21,11)(13,24,25,16)(14,23,27,22)(15,29,31,17)(20,28,32,30);(1,4)(2,9)(3,13)(5,20)(6,24)(7,16)(8,18)(10,26)(11,28)(12,25)(14,15)(17,23)(19,31)(21,29)(22,32)(27,30);(2,10)(3,12)(4,16)(5,21)(9,25)(11,14)(13,26)(17,29)(18,24)(19,27)(20,32)(22,23)

Polycyclic presentation:

2^10|b^2=g,e^2=h,b^a=bg,c^a=ce,c^b=cf,d^a=dghij,d^b=dj,d^c=dh,e^a=eh,e^b=ei,e^c=eh,e^d=ehj,f^a=fhij,f^b=fh,f^e=fj,g^c=gh,i^c=ij

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72262804, #72262833
maximal normal subgroup of#72262804, #72262833

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