#6629532

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	#60426	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2200 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|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, 6536728);