#6583122

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574270
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3098 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ehi,d^c=dhi,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6490318);