#6357658

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357620
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

1885 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264854);