#6486332

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1380 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=h,e^2=fgi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei

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, 6393528);