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#72318052

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#139716
commutatorC23×Q8C22×C4
Frattini, cosocleC2×#536C23
prime radical#449493C2
LCSG ▷ C23×Q8 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #35616 ◁ G
LECSG ▷ C2×#536 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

2031 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 2031 conjugacy classes of subgroups — too many to draw. Only its 77 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:

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

Polycyclic presentation:

2^10|b^2=d,c^2=j,f^2=j,g^2=j,h^2=j,c^a=ce,c^b=cf,d^c=dg,e^b=eghj,e^d=ej,f^a=fghj,f^b=fg,f^c=fj,f^d=fij,g^a=gj,g^b=gi,g^c=gj,g^d=gj,g^f=gj,h^a=hj,h^b=hij,h^c=hj,h^d=hj,h^f=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

49 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72318072, #72318106, #72318208, …
maximal normal subgroup of#72318072, #72318106, #72318208, …

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