#6607315

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1388 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)(9,10,12,15)(11,14,16,13)(17,18)(19,24)(20,29)(21,25)(22,31)(23,26)(27,32)(28,30);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,16)(14,15)(17,19,23,22)(18,24,26,31)(20,25,30,27)(21,28,32,29);(11,16)(13,14)(17,20,19,27,23,30,22,25)(18,21,31,28,26,32,24,29);(9,11)(10,14)(12,16)(13,15)(17,21,23,32)(18,25,26,27)(19,28,22,29)(20,24,30,31);(11,16)(13,14)(17,19,23,22)(18,24,26,31)(20,27,30,25)(21,29,32,28)

Polycyclic presentation:

2^9|a^2=f,b^2=fh,c^2=gi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,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, 6514511);