#6623955

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	C₂×#459	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1518 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)(17,18,21,24,22,26,20,25)(19,27,29,31,28,32,30,23);(2,5)(6,8)(10,14)(12,15)(17,19,22,28)(18,23,26,31)(20,29,21,30)(24,32,25,27);(9,10)(11,12)(13,14)(15,16)(17,18,22,26)(19,23,28,31)(20,24,21,25)(27,29,32,30);(1,3,4,7)(2,6,5,8)(9,11,13,16)(10,15,14,12)(19,28)(23,31)(27,32)(29,30);(9,12,13,15)(10,11,14,16)(17,20,22,21)(18,24,26,25)(19,29,28,30)(23,32,31,27)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,d^2=fgh,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,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, 6531151);