#504462

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
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	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

636 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)(13,14,19,31,30,39,29,20,18,25,16,23,17,24,32,22)(15,21,33,42,41,44,40,34,28,38,26,36,27,37,43,35);(1,2)(3,4)(14,20)(16,29)(17,30)(19,32)(21,34)(22,25)(23,24)(26,40)(27,41)(31,39)(33,43)(35,38)(36,37)(42,44);(2,4)(6,9)(10,12)(13,15,18,28)(14,21,25,38)(16,26,19,33)(17,27,30,41)(20,34,22,35)(23,36,31,42)(24,37,39,44)(29,40,32,43);(5,7,8,11)(6,10,9,12)(14,22)(16,29)(17,30)(19,32)(20,25)(21,35)(23,39)(24,31)(26,40)(27,41)(33,43)(34,38)(36,44)(37,42)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411658);