Finite Group Databasebeta

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#72287337

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#130204
commutatorC2×M32C22×C4
Frattini, cosocleC2×#504C23
prime radical#7624857C2
LCSG ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ #33434 ◁ G
LECSG ▷ C2×#504 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

873 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 873 conjugacy classes of subgroups — too many to draw. Only its 111 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=g,c^2=d,d^2=hi,e^2=h,f^2=j,h^2=ij,i^2=j,c^a=cf,c^b=ce,d^b=dh,e^a=eg,e^b=ehij,f^b=fgj,f^c=fj,f^e=fj,g^c=gj,h^b=hi,i^b=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72287373, #72321146, #72342920
maximal normal subgroup of#72287373, #72321146, #72342920

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