#504268

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

524 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=f,c^2=fh,d^2=eh,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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411464);