#499482

order	512
nilpotency class	4
isoclinism	isoclinic class of #13304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂³	C₂²×D₈
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

630 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)(7,10)(8,9)(11,12)(13,14,16,20,28,35,41,39,18,22,30,37,17,21,29,36)(15,23,24,38,31,44,43,33,26,40,42,32,25,34,27,19);(1,2)(3,4)(6,9)(10,12);(2,4)(13,15,18,26)(14,19,22,33)(16,27,30,43)(17,31,28,25)(20,34,37,44)(21,38,35,32)(23,39,40,36)(24,41,42,29);(5,7,8,11)(6,10,9,12)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,f^c=fh,h^c=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406678);