#509284

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
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

594 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)(13,14,16,20)(15,23,25,19)(17,21,28,35)(18,22,29,36)(24,39,41,32)(26,40,42,33)(27,38,31,34)(30,37,43,44);(5,7)(6,10)(8,11)(9,12)(13,15,16,25)(14,19,20,23)(17,24,28,41)(18,31,29,27)(21,32,35,39)(22,38,36,34)(26,43,42,30)(33,44,40,37);(2,4)(5,8,9,10)(6,7,11,12)(15,24)(17,30)(18,29)(19,32)(21,37)(22,36)(23,39)(25,41)(26,27)(28,43)(31,42)(33,34)(35,44)(38,40);(2,4)(6,11)(7,12)(15,25)(19,23)(24,41)(26,42)(27,31)(32,39)(33,40)(34,38)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=ei,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,g^c=gh,h^c=hi

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, 416480);