#512444

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	#33308	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

785 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)(14,20)(15,25)(16,17)(18,29)(21,32)(23,34)(24,36)(26,27)(28,40)(30,31)(33,37)(38,43)(39,41)(42,44);(6,9)(10,12)(13,14,17,23,18,24,30,32,19,21,31,36,29,34,16,20)(15,22,25,37,27,38,39,42,28,35,40,44,41,43,26,33);(1,2,3,4)(14,21)(20,32)(22,35)(23,36)(24,34)(33,42)(37,44)(38,43);(5,7,8,11)(6,10,9,12)(13,15,16,26,29,41,31,40,19,28,30,39,18,27,17,25)(14,22,20,33,34,43,36,44,21,35,32,42,24,38,23,37)

Polycyclic presentation:

2^9|b^2=g,c^2=f,d^2=e,e^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,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, 419640);