#503769

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

534 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)(15,25)(16,30)(17,33)(18,23)(19,36)(20,28)(21,32)(22,35)(24,34)(26,39)(27,41)(29,44)(31,37)(38,42)(40,43);(1,2)(3,4)(13,15,26,32,19,29,17,28,18,27,35,43,34,16,31,42)(14,20,37,44,24,21,22,25,23,38,33,30,36,40,39,41);(2,4)(5,7,8,12)(6,9,10,11)(15,27)(16,29)(20,38)(21,40)(25,41)(28,42)(30,44)(32,43);(13,16,26,42,19,15,17,32,18,29,35,28,34,27,31,43)(14,21,37,25,24,38,22,30,23,40,33,41,36,20,39,44)

Polycyclic presentation:

2^9|a^2=f,b^2=ehi,c^2=f,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,d^c=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, 410965);