#6353241

order	512
nilpotency class	3
isoclinism	isoclinic class of #6353061
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	#62892	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3077 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6260437);