#509259

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

589 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2)(3,4)(5,6)(7,8)(9,11)(10,12);(5,7)(6,10)(8,11)(9,12)(13,14,19,23)(15,24,27,42)(16,21,18,33)(17,29,32,43)(20,35,38,26)(22,28,40,31)(25,41,34,39)(30,36,44,37);(2,4)(5,8,9,10)(6,7,11,12)(13,15)(14,20)(16,25)(17,30)(18,34)(19,27)(21,36)(22,39)(23,38)(24,29)(26,31)(28,35)(32,44)(33,37)(40,41)(42,43);(2,4)(6,11)(7,12)(13,16,19,18)(14,21,23,33)(15,25,27,34)(17,28,32,31)(20,36,38,37)(22,29,40,43)(24,41,42,39)(26,30,35,44)

Polycyclic presentation:

2^9|b^2=i,c^2=e,d^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,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416455);