#502002

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	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

552 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)(7,9)(8,10)(11,12)(13,14,17,20)(15,23,26,37)(16,28,30,43)(18,31,29,22)(19,32,33,25)(21,36,35,24)(27,42,41,39)(34,40,38,44);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,15,24,40,18,27,30,32,17,26,36,44,29,41,16,25)(14,19,28,39,22,34,35,37,20,33,43,42,31,38,21,23);(2,4)(6,10)(9,12)(14,20)(19,33)(21,35)(22,31)(23,37)(28,43)(34,38)(39,42);(13,15,24,40,18,27,30,32,17,26,36,44,29,41,16,25)(14,19,28,39,22,34,35,37,20,33,43,42,31,38,21,23)

Polycyclic presentation:

2^9|a^2=i,b^2=efhi,d^2=eh,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg

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, 409198);