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#72225499

order2048
nilpotency class6
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72225497
commutatorC4×C32C22×C4
Frattini, cosocleC4×M64C23
prime radical#72225424C2
LCSG ▷ C4×C32 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C4×C8 ◁ C4×#346 ◁ C4×#1398 ◁ G
LECSG ▷ C4×M64 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

684 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 64
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,7,11)(3,8,18,30)(4,21,22,57)(5,25,27,34)(6,9,26,12)(10,35,36,15)(13,43,39,17)(14,44,40,23)(16,28,20,41)(19,45,54,64)(24,58,52,56)(29,60,61,48)(31,51,62,47)(32,50,37,46)(33,49,42,53)(38,63,59,55);(2,8)(3,15)(4,17)(5,23)(6,26)(9,28)(10,13)(11,30)(12,41)(14,45)(16,46)(18,35)(19,55)(20,50)(21,29)(22,43)(24,49)(25,31)(27,44)(32,42)(33,37)(34,62)(36,39)(38,56)(40,64)(47,48)(51,60)(52,53)(54,63)(57,61)(58,59);(2,9,11,12)(3,16,18,20)(4,22)(10,36)(13,42,39,33)(14,34,40,25)(15,46,35,50)(17,43)(19,51,54,47)(21,56,57,58)(24,52)(29,61)(32,37)(38,59)(48,55,60,63)(49,53)

Polycyclic presentation:

2^11|a^2=k,c^2=e,d^2=gi,f^2=h,g^2=ij,i^2=jk,j^2=k,b^a=bd,c^a=cj,c^b=cf,d^a=dg,d^b=dg,d^c=djk,e^a=ek,e^b=eh,e^d=ek,f^a=fjk,f^b=fh,g^a=gi,g^b=gi,g^c=gk,h^a=hk,i^a=ij,i^b=ij,j^a=jk,j^b=jk

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72225510
maximal normal subgroup of#72225510

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.11);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.7*F.9);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 9, F.10*F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.10*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.10*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);