#6401878

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×#120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1397 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)(3,5)(4,6)(7,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26,30,34,29,31,27,32)(28,36,38,39,37,33,35,40);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(26,31)(32,34)(33,36)(39,40);(2,6)(5,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,27,29,30)(26,32,31,34)(28,35,37,38)(33,39,36,40);(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,27,29,30)(26,32,31,34)(28,35,37,38)(33,39,36,40);(10,18)(15,19)(16,17)(23,24)(25,28,30,38,29,37,27,35)(26,33,34,40,31,36,32,39)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=f,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,e^a=ef,d^b=dfi

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6309074);