#6550365

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533087
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1490 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)(3,6)(4,5)(7,8)(25,26,28,32,29,33,36,31)(27,30,34,38,35,39,40,37);(2,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22)(25,27,29,35)(26,30,33,39)(28,34,36,40)(31,37,32,38);(1,3,4,7)(2,6,5,8)(10,14)(15,21)(16,22)(23,24)(26,31)(28,36)(30,37)(32,33)(34,40)(38,39);(9,11,13,18)(10,15,14,21)(12,19,20,17)(16,24,22,23);(9,12,13,20)(10,16,14,22)(11,17,18,19)(15,23,21,24)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6457561);