#6499038

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470867
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1286 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,19)(13,15)(16,20)(17,21)(18,24)(22,23)(25,26,30,32)(27,34,37,31)(28,38,33,29)(35,36,39,40);(2,6)(7,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,22)(19,24)(20,23)(25,27)(26,31)(28,35)(29,36)(30,37)(32,34)(33,39)(38,40);(1,3,5,4)(2,7,6,8)(10,15)(11,17)(16,19)(18,23)(25,28,29,26,30,33,38,32)(27,35,36,31,37,39,40,34);(1,4,5,3)(2,8,6,7)(11,17)(14,21)(18,23)(22,24)(25,29,30,38)(26,28,32,33)(27,36,37,40)(31,35,34,39);(9,12,13,20)(10,16,15,19)(11,18,17,23)(14,22,21,24)(27,37)(31,34)(35,39)(36,40)

Polycyclic presentation:

2^9|a^2=i,c^2=fghi,d^2=f,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^b=ei,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6406234);