#502012

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	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

578 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)(13,14,19,24)(15,25,16,29)(17,33,35,38)(18,36,34,23)(20,31,21,27)(22,30,40,26)(28,44,32,41)(37,42,39,43);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,15,26,42,18,28,35,31,19,16,30,43,34,32,17,27)(14,20,33,41,23,37,40,29,24,21,38,44,36,39,22,25);(2,4)(6,10)(9,12);(13,16,30,42,18,32,17,31,19,15,26,43,34,28,35,27)(14,21,38,41,23,39,22,29,24,20,33,44,36,37,40,25)

Polycyclic presentation:

2^9|a^2=i,b^2=efh,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, 409208);