#6358006

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357957
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

1851 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=g,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,e^b=ei,d^c=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, 6265202);