#509775

order	512
nilpotency class	4
isoclinism	isoclinic class of #509163
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

709 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,7)(4,6)(5,8)(9,10,14,19,13,18,31,15)(11,20,24,21,23,16,25,33)(12,26,29,35,28,36,30,17)(22,38,39,32,27,37,40,34);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,27)(13,23)(14,24)(15,21)(17,34)(18,20)(19,33)(22,28)(25,31)(26,38)(29,40)(30,39)(32,35)(36,37);(10,15)(11,21)(14,31)(16,25)(17,26)(18,19)(20,24)(22,37)(23,33)(27,38)(29,30)(32,39)(34,40)(35,36);(2,6)(3,8)(9,12)(10,17)(11,22)(13,28)(14,30)(15,26)(16,32)(18,35)(19,36)(20,34)(21,37)(23,27)(24,40)(25,39)(29,31)(33,38)

Polycyclic presentation:

2^9|a^2=f,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defi,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416971);