#6996396

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996395
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

2271 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=f,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=dh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6903592);