#6623994

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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1480 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)(3,6)(4,5)(7,8)(17,18,23,29,24,30,22,28)(19,26,20,25,31,21,27,32);(2,5)(6,8)(10,14)(12,15)(17,19,24,31)(18,25,30,32)(20,23,27,22)(21,29,26,28);(9,10)(11,12)(13,14)(15,16)(17,20,24,27)(18,26,30,21)(19,22,31,23)(25,29,32,28);(1,3,4,7)(2,6,5,8)(9,11,13,16)(10,15,14,12)(17,21)(18,27)(19,29)(20,30)(22,25)(23,32)(24,26)(28,31);(9,12,13,15)(10,11,14,16)(17,22,24,23)(18,28,30,29)(19,20,31,27)(21,32,26,25)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=fghi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,f^b=fi,g^b=gi,d^c=dh,f^c=fi,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, 6531190);