#6565219

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1384 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,d^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^c=di,e^d=eh

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, 6472415);