#6350198

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350061
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

2832 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,f^b=fh,g^b=gi,f^c=fi,g^c=gi,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, 6257394);