#6411997

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
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×#41	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1499 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)(11,18)(12,16)(13,22)(14,17)(15,20)(19,24)(21,23)(25,26,30,31,29,33,27,34)(28,36,38,40,37,32,35,39);(1,3)(2,5)(4,7)(6,8)(9,11,14,20)(10,15,17,18)(12,19,13,21)(16,23,22,24)(25,27,29,30)(26,31,33,34)(28,35,37,38)(32,39,36,40);(2,6)(3,7)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,28,27,35,29,37,30,38)(26,32,34,40,33,36,31,39);(2,6)(5,8)(11,20)(15,18)(19,21)(23,24);(9,13,14,12)(10,16,17,22)(11,21,20,19)(15,23,18,24)(25,27,29,30)(26,31,33,34)(28,35,37,38)(32,39,36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6319193);