#6358061

order	512
nilpotency class	3
isoclinism	isoclinic class of #6358057
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

1968 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)(3,9)(4,8)(5,10)(6,7)(11,15)(12,14)(13,16)(19,25)(20,27)(22,28)(24,30)(26,31)(29,32);(1,3,9,2,6,10,5,7)(4,11,15,8,12,16,13,14)(17,18)(19,22)(20,24)(21,23)(25,28)(26,29)(27,30)(31,32);(2,7)(3,10)(4,12)(13,15)(17,19,20,25,21,26,27,31)(18,22,24,28,23,29,30,32);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(19,26)(22,29)(25,31)(28,32);(4,12)(8,14)(11,16)(13,15)(18,23)(22,29)(24,30)(28,32)

Polycyclic presentation:

2^9|b^2=fh,c^2=gh,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,e^b=ei,d^c=dhi,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, 6265257);