#509774

order	512
nilpotency class	4
isoclinism	isoclinic class of #509155
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

682 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,h^b=hi,f^c=fi,g^c=gh,h^c=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416970);