#6597324

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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	C₂×#546	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1792 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6504520);