#6358086

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

1866 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=gh,d^2=h,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, 6265282);