#512459

order	512
nilpotency class	4
isoclinism	isoclinic class of #511951
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2250	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

693 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=fh,d^2=e,e^2=hi,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,g^a=ghi,h^a=hi,c^b=ci

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419655);