#6351901

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351885
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2812 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259097);