#6604525

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1372 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=f,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,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

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6511721);