#509806

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
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

705 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)(13,14,17,21,32,38,43,42,19,23,34,40,18,22,33,39)(15,24,25,41,29,44,31,36,16,27,28,35,26,37,30,20);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11);(2,4)(13,15,19,16)(14,20,23,36)(17,30,34,31)(18,29,32,26)(21,37,40,44)(22,41,38,35)(24,42,27,39)(25,43,28,33);(6,10)(7,12)(13,16,19,15)(14,20,23,36)(17,31,34,30)(18,26,32,29)(21,37,40,44)(22,41,38,35)(24,42,27,39)(25,33,28,43)

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=defi,c^b=cg,d^b=de,f^c=fh,h^c=hi,f^d=fh,h^d=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, 417002);