#502335

order	512
nilpotency class	4
isoclinism	isoclinic class of #500834
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 ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

542 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)(7,11)(8,9)(10,12)(13,14,18,24,36,41,33,43,17,23,19,25,20,26,37,42)(15,27,29,22,16,32,34,44,28,21,30,38,31,39,35,40);(1,2)(3,4)(6,9)(10,11)(13,15,18,29,36,16,33,34,17,28,19,30,20,31,37,35)(14,21,24,38,41,39,43,40,23,27,25,22,26,32,42,44);(2,4)(13,16,17,31)(14,22,23,38)(15,20,28,36)(18,29,19,30)(21,24,27,25)(26,44,41,40)(32,43,39,42)(33,35,37,34);(5,7,8,12)(6,10,9,11)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=ei,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fh,g^c=gh,h^c=hi

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, 409531);