#6498990

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470884
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1230 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=gh,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,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, 6406186);