#512395

order	512
nilpotency class	4
isoclinism	isoclinic class of #511951
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

783 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 419591);