#502387

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
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

584 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=i,d^2=ei,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh,g^c=gh,h^c=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409583);