#6612637

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#24397	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1738 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=ehi,d^c=dhi,e^c=ei,e^d=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6519833);