#6608867

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	#25115	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1511 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)(3,7)(4,8)(5,6)(9,10)(11,16)(12,14)(13,15)(17,18,23,19)(20,28,31,32)(21,25,22,27)(24,30,26,29);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19,23,18)(20,24,31,26)(21,25,22,27)(28,30,32,29);(1,3,5,4)(2,7,6,8)(10,15)(11,12)(17,20,22,29,23,31,21,30)(18,24,25,28,19,26,27,32);(1,4,5,3)(2,8,6,7)(9,12,13,11)(10,14,15,16)(17,21,23,22)(18,25,19,27)(20,29,31,30)(24,32,26,28);(11,12)(14,16)(17,21,23,22)(18,25,19,27)(20,30,31,29)(24,28,26,32)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=fg,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,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, 6516063);