#6352563

order	512
nilpotency class	3
isoclinism	isoclinic class of #6352469
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
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2788 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259759);