#7611881

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563333
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1882 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 72 points:

(1,2)(3,5)(4,6)(7,8)(9,10)(11,24)(12,29)(13,21)(14,40)(15,41)(16,23)(17,22)(18,27)(19,34)(20,39)(25,56)(26,51)(28,47)(30,54)(31,49)(32,65)(33,50)(35,61)(36,55)(37,52)(38,53)(42,46)(43,62)(44,66)(45,60)(48,59)(57,72)(58,70)(63,71)(64,68)(67,69);(1,3)(2,5)(4,7)(6,8)(9,11,15,27,16,28,17,42)(10,18,22,24,23,46,41,47)(12,30,33,62,34,25,35,59)(13,36,38,60,39,66,14,26)(19,48,50,56,29,43,61,54)(20,51,53,44,21,45,40,55)(31,63,32,57,37,67,64,58)(49,72,68,71,52,70,65,69);(2,6)(3,7)(9,12)(10,19)(11,25)(13,37)(14,32)(15,33)(16,34)(17,35)(18,43)(20,52)(21,49)(22,50)(23,29)(24,54)(26,58)(27,59)(28,30)(31,39)(36,63)(38,64)(40,68)(41,61)(42,62)(44,71)(45,70)(46,48)(47,56)(51,72)(53,65)(55,69)(57,60)(66,67);(2,6)(5,8)(9,13,17,14,16,39,15,38)(10,20,22,53,23,21,41,40)(11,26,27,36,28,60,42,66)(12,31,35,64,34,37,33,32)(18,44,47,51,46,55,24,45)(19,49,50,68,29,52,61,65)(25,57,59,67,30,58,62,63)(43,69,56,70,48,71,54,72);(9,14,17,39,16,38,15,13)(10,21,41,53,23,20,22,40)(11,26,42,66,28,60,27,36)(12,32,35,31,34,64,33,37)(18,45,47,44,46,51,24,55)(19,49,61,65,29,52,50,68)(25,58,62,67,30,57,59,63)(43,70,56,71,48,72,54,69)

Polycyclic presentation:

2^9|b^2=f,d^2=h,e^2=fi,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7519077);