#502010

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
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

550 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)(15,23,27,38)(16,29,31,34)(18,32,30,22)(19,33,35,26)(21,24,37,25)(28,44,43,40)(36,42,39,41);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,15,24,41,18,28,16,26,17,27,25,42,30,43,31,33)(14,19,34,44,22,36,21,23,20,35,29,40,32,39,37,38);(2,4)(6,10)(9,12)(14,20)(19,35)(21,37)(22,32)(23,38)(29,34)(36,39)(40,44);(13,15,25,42,18,28,31,33,17,27,24,41,30,43,16,26)(14,19,29,40,22,36,37,38,20,35,34,44,32,39,21,23)

Polycyclic presentation:

2^9|a^2=i,b^2=efh,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,d^b=di

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, 409206);