#509630

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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 ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

612 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)(14,19)(15,23)(21,35)(22,32)(25,40)(26,33)(29,41)(30,44);(6,9)(7,12)(13,14)(15,24)(16,19)(17,29)(18,32)(20,25)(21,31)(22,28)(23,39)(26,43)(27,41)(30,38)(33,37)(34,40)(35,42)(36,44);(2,4)(13,15,16,23)(14,20,19,34)(17,30,27,44)(18,33,28,26)(21,37,35,43)(22,38,32,36)(24,41,39,29)(25,42,40,31);(2,4)(5,7,8,12)(6,10,9,11)(14,19)(20,34)(21,35)(22,32)(24,39)(29,41)(36,38)(37,43)

Polycyclic presentation:

2^9|a^2=ei,c^2=i,d^2=ei,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,h^b=hi,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, 416826);