#6357935

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357891
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1822 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(17,18)(19,27)(20,25)(21,33)(22,24)(23,30)(26,34)(28,40)(29,39)(31,37)(32,45)(35,43)(36,41)(38,44)(42,47)(46,48);(17,19,21,30,22,29,34,43)(18,23,26,27,24,35,33,39)(20,28,32,41,31,42,44,48)(25,36,38,40,37,46,45,47);(1,3,5,13,6,11,4,12)(2,7,8,10,9,15,14,16)(18,24)(23,35)(25,37)(26,33)(27,39)(36,46)(38,45)(40,47);(3,11)(7,15)(10,16)(12,13)(17,20,22,31)(18,25,24,37)(19,28,29,42)(21,32,34,44)(23,36,35,46)(26,38,33,45)(27,40,39,47)(30,41,43,48);(1,4,6,5)(2,8,9,14)(3,12,11,13)(7,10,15,16)(19,29)(20,31)(23,35)(25,37)(27,39)(30,43)(32,44)(38,45)

Polycyclic presentation:

2^9|b^2=f,c^2=gh,d^2=h,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,e^b=eh,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, 6265131);