#6353333

order	512
nilpotency class	3
isoclinism	isoclinic class of #6353301
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

2821 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6260529);