#6996408

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996405
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2213 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,5)(4,6)(7,8)(9,10,14,20,16,21,15,32)(11,22,24,37,25,36,27,17)(12,26,28,18,29,38,33,35)(13,19,30,34,31,39,23,40);(1,2)(3,7)(4,6)(5,8)(9,11,16,25)(10,17,21,37)(12,23,29,30)(13,28,31,33)(14,27,15,24)(18,34,35,40)(19,26,39,38)(20,36,32,22);(1,3)(2,5)(4,8)(6,7)(9,12)(10,18)(11,23)(13,24)(14,28)(15,33)(16,29)(17,34)(19,36)(20,38)(21,35)(22,39)(25,30)(26,32)(27,31)(37,40);(3,8)(5,7)(9,13,15,23,16,31,14,30)(10,19,32,40,21,39,20,34)(11,12,27,33,25,29,24,28)(17,35,36,38,37,18,22,26);(2,6)(5,7)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=fi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6903604);