#6611278

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1490 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:

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

Polycyclic presentation:

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

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, 6518474);