#6629533

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629034
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2152 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)(9,10,16,20)(11,21,23,32)(12,14,19,15)(13,26,28,34)(17,25,31,24)(18,29,33,27)(22,36,37,30)(35,39,40,38);(2,5)(6,8)(9,11,14,24,16,23,15,25)(10,17,12,21,20,31,19,32)(13,22,27,38,28,37,29,39)(18,30,26,35,33,36,34,40);(1,3,4,7)(2,6,5,8)(9,12,14,20,16,19,15,10)(11,21,24,31,23,32,25,17)(13,26,27,33,28,34,29,18)(22,35,38,36,37,40,39,30);(9,13)(10,18)(11,22)(12,26)(14,27)(15,29)(16,28)(17,30)(19,34)(20,33)(21,35)(23,37)(24,38)(25,39)(31,36)(32,40);(11,23)(17,31)(21,32)(22,37)(24,25)(30,36)(35,40)(38,39)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,h^a=hi,e^b=ei,h^d=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536729);