#6603088

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1410 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:

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

Polycyclic presentation:

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

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, 6510284);