#509790

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

708 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,f^c=fh,h^c=hi,f^d=fh,h^d=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, 416986);