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#72225585

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#123275
commutator, Frattini, cosocleC8×C16C23
prime radical#492533C2
LCSG ▷ C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ C8×C16 ◁ G
LECSG ▷ C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

359 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 359 conjugacy classes of subgroups — too many to draw. Only its 55 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:

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

Polycyclic presentation:

2^10|a^2=j,b^2=j,c^2=df,d^2=fh,e^2=gi,f^2=h,g^2=ij,i^2=j,b^a=be,c^a=cej,c^b=cd,d^a=dg,d^b=df,e^a=eg,e^b=eg,f^a=fi,f^b=fh,g^a=gi,g^b=gi,h^a=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72225587, #72225590
maximal normal subgroup of#72225587, #72225590

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