#512460

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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

651 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)(13,14,22,29)(15,19,27,18)(16,32,37,42)(17,38,39,30)(20,26,41,25)(21,31,40,28)(23,34,43,35)(24,44,33,36);(6,9)(10,12)(13,15,20,31,21,26,18,14,22,27,41,28,40,25,19,29)(16,33,17,30,36,42,34,43,37,24,39,38,44,32,35,23);(1,2,3,4)(13,16,20,17,21,36,18,34,22,37,41,39,40,44,19,35)(14,23,27,33,28,30,25,42,29,43,15,24,31,38,26,32);(5,7,8,11)(6,10,9,12)(13,17,18,37,40,35,20,36,22,39,19,16,21,34,41,44)(14,24,25,23,31,42,27,38,29,33,26,43,28,32,15,30)

Polycyclic presentation:

2^9|a^2=i,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, 419656);