#6550580

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533105
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1458 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)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26,30,33,29,31,28,37)(27,34,35,39,36,40,38,32);(2,6)(7,8)(10,14)(11,17)(16,21)(18,23)(26,31)(32,39)(33,37)(34,40);(1,3,5,4)(2,7,6,8)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,27,29,36)(26,32,31,39)(28,35,30,38)(33,40,37,34);(1,4,5,3)(2,8,6,7)(12,19)(16,21)(18,23)(22,24);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,28,29,30)(26,33,31,37)(27,35,36,38)(32,40,39,34)

Polycyclic presentation:

2^9|a^2=g,c^2=f,d^2=fhi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6457776);