Finite Group Databasebeta

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#72225475

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72225474
commutatorC4×C32C22×C4
Frattini, cosocle#7329C23
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 ▷ #7329 ▷ 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:

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

Polycyclic presentation:

2^11|a^2=e,c^2=gik,d^2=fghk,e^2=gijk,f^2=hjk,g^2=ik,h^2=jk,i^2=k,j^2=k,b^a=bd,c^a=cjk,c^b=cgk,d^a=df,d^b=dfgi,e^b=egi,e^d=ek,f^a=fh,f^b=fhk,g^a=gk,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#72225481, #72225515, #72225523
maximal normal subgroup of#72225481, #72225515, #72225523

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