#509939

order	512
nilpotency class	4
isoclinism	isoclinic class of #509163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#25434	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

535 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)(3,5)(4,6)(7,8)(9,10,16,19,17,20,14,15)(11,13,30,33,23,32,22,31)(12,24,28,18,29,34,27,35)(21,36,39,26,38,25,37,40);(1,3,5,2)(4,7,8,6)(9,11)(10,13)(12,25)(14,22)(15,31)(16,30)(17,23)(18,21)(19,33)(20,32)(24,37)(26,27)(28,40)(29,36)(34,39)(35,38);(1,4,5,8)(2,6,3,7)(9,12)(10,18)(11,21)(13,26)(14,28)(15,34)(16,27)(17,29)(19,24)(20,35)(22,39)(23,38)(25,31)(30,37)(32,40)(33,36);(2,3)(6,7)(9,13)(10,11)(12,26)(14,33)(15,30)(16,31)(17,32)(18,21)(19,22)(20,23)(24,39)(25,27)(28,36)(29,40)(34,37)(35,38)

Polycyclic presentation:

2^9|a^2=fi,b^2=e,c^2=e,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, 417135);