Finite Group Databasebeta

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#72225436

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72225435
commutatorC4×C32C22×C4
Frattini, cosocleC8×C32C23
prime radical#72216261C2
LCSG ▷ C4×C32 ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C4×C8 ◁ C2×C8×C16 ◁ C8×M64 ◁ G
LECSG ▷ C8×C32 ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

495 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 495 conjugacy classes of subgroups — too many to draw. Only its 104 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:

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

Polycyclic presentation:

2^11|a^2=e,d^2=fgh,e^2=g,f^2=hj,g^2=ik,h^2=jk,i^2=k,j^2=k,b^a=bd,c^a=ck,c^b=cijk,d^a=df,d^b=dfgi,d^c=dk,e^b=egi,f^a=fh,f^b=fh,g^b=gi,h^a=hj,h^b=hj,i^b=ik,j^a=jk,j^b=jk

Conjugacy classes & character table

134 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72225451, #72225523, #72225534
maximal normal subgroup of#72225451, #72225523, #72225534

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