#6499048

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470874
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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1358 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)(25,26,29,32)(27,33,34,40)(28,38,35,31)(30,36,39,37);(2,6)(7,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,22)(19,24)(20,23)(27,34)(30,39)(33,40)(36,37);(1,3,5,4)(2,7,6,8)(10,15)(11,17)(16,19)(18,23)(25,27,35,36,29,34,28,37)(26,30,38,40,32,39,31,33);(1,4,5,3)(2,8,6,7)(11,17)(14,21)(18,23)(22,24);(9,12,13,20)(10,16,15,19)(11,18,17,23)(14,22,21,24)(25,28,29,35)(26,31,32,38)(27,36,34,37)(30,40,39,33)

Polycyclic presentation:

2^9|a^2=i,c^2=fgh,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=dhi,e^c=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, 6406244);