#72306604

order	1024
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#13561
commutator	#127	C₂³×C₄
Frattini, cosocle	#493	C₂⁴
prime radical	#60639	C₂²
LCS	G ▷ #127 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄² ◁ #60639 ◁ G
LECS	G ▷ #493 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

1831 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^10|b^2=ghi,c^2=e,e^2=hij,g^2=i,h^2=j,b^a=bij,c^a=cj,c^b=cf,d^a=dj,d^b=dh,d^c=dgj,e^b=eg,e^d=ehij,f^a=fj,f^b=fhj,f^c=fgi,f^e=fhij,g^b=gi,g^c=gh,g^d=gi,g^f=gi,h^b=hi,h^c=hj,h^d=hj,h^f=hj,i^c=ij

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306606, #72306617, #72306761, …
maximal normal subgroup of	#72306606, #72306617, #72306761, …

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