#6629536

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629032
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	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2208 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)(9,10)(11,21)(12,16)(13,27)(14,31)(15,20)(17,24)(18,29)(19,22)(23,32)(25,34)(26,33)(28,36)(30,35)(37,39)(38,40);(2,5)(6,8)(9,11,22,26,15,24,14,12)(10,16,31,17,20,33,19,21)(13,23,29,39,30,38,28,25)(18,32,27,34,36,40,35,37);(1,3,4,7)(2,6,5,8)(9,12,14,24,15,26,22,11)(10,17,19,16,20,21,31,33)(13,25,28,38,30,39,29,23)(18,34,35,32,36,37,27,40);(9,13)(10,18)(11,23)(12,25)(14,28)(15,30)(16,32)(17,34)(19,35)(20,36)(21,37)(22,29)(24,38)(26,39)(27,31)(33,40);(11,24)(12,26)(16,33)(17,21)(23,38)(25,39)(32,40)(34,37)

Polycyclic presentation:

2^9|b^2=gi,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,e^c=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, 6536732);