#504281

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	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

566 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=fhi,d^2=ehi,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,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, 411477);