#430892

order	512
nilpotency class	3
isoclinism	isoclinic class of #418438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1145 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|b^2=ei,c^2=f,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,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, 338088);