#6352689

order	512
nilpotency class	3
isoclinism	isoclinic class of #6352581
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	C₂²×C₄	C₂⁵
prime radical	#62930	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2951 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259885);