#503765

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

530 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eh,c^2=fi,d^2=ehi,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, 410961);