#506315

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	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

538 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=gi,d^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,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, 413511);