Finite Group Databasebeta

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#72226942

order2048
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Subgroups

subgroupG/H
centerC4#123285
commutatorC8×C16C22×C4
Frattini, cosocleC2×C8×C16C23
socleC2#72225202
prime radical#491465C22
LCSG ▷ C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C8×C16 ◁ G
LECSG ▷ C2×C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

740 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

Polycyclic presentation:

2^11|a^2=e,c^2=d,d^2=gik,e^2=ghi,f^2=hk,g^2=i,h^2=jk,j^2=k,b^a=befhjk,c^a=cfik,c^b=cefghi,d^a=dh,d^b=dg,e^b=eghj,e^c=ek,f^a=fhjk,f^b=fhjk,f^c=fk,g^a=gj,g^b=gi,h^a=hj,h^b=hj,i^a=ik,j^a=jk,j^b=jk

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72226950, #72226961
maximal normal subgroup of#72226950, #72226961

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.5);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 4, F.7*F.9*F.11);;
gap> SetPower(coll, 5, F.7*F.8*F.9);;
gap> SetPower(coll, 6, F.8*F.11);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10*F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.5*F.6*F.8*F.10*F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.9*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.10*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.10*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
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);