#6357239

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357223
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

1900 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)(4,9)(5,7)(6,14)(8,13)(11,15)(12,16)(17,18,23,19)(20,22,27,24)(21,25,29,26)(28,30,32,31);(2,7)(8,15)(9,14)(10,16)(17,19,22,27,23,18,24,20)(21,26,30,32,29,25,31,28);(1,3,6,13,5,12,4,11)(2,8,9,10,7,15,14,16)(17,20,22,19,23,27,24,18)(21,28,30,26,29,32,31,25);(1,4,5,6)(2,9,7,14)(3,11,12,13)(8,10,15,16)(17,21,23,29)(18,25,19,26)(20,28,27,32)(22,30,24,31);(21,29)(25,26)(28,32)(30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264435);