#512445

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

727 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)(13,14,20,25)(15,26,27,28)(16,32,22,18)(17,33,35,23)(19,36,34,24)(21,37,38,29)(30,40,43,41)(31,39,44,42);(6,9)(10,12)(13,15,18,30,19,31,17,29,20,27,32,43,34,44,35,37)(14,21,23,39,24,40,22,26,25,38,33,42,36,41,16,28);(1,2,3,4)(15,27)(21,38)(26,28)(29,37)(30,43)(31,44)(39,42)(40,41);(5,7,8,11)(6,10,9,12)(13,16,17,24,34,33,18,14,20,22,35,36,19,23,32,25)(15,28,29,40,44,42,30,21,27,26,37,41,31,39,43,38)

Polycyclic presentation:

2^9|a^2=i,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, 419641);