#6548908

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1336 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=i,d^2=h,e^2=h,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6456104);