#7007227

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

1944 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)(11,23,27,39)(12,28,25,40)(13,31)(14,20)(15,21)(16,22)(17,26,36,29)(18,30,37,32)(19,34)(24,35)(33,38);(1,2)(3,7)(4,6)(5,8)(9,11)(10,17)(12,24)(13,32)(14,29)(15,26)(16,27)(18,35)(19,28)(20,39)(21,23)(22,36)(25,33)(30,34)(31,40)(37,38);(1,3)(2,5)(4,8)(6,7)(9,12,16,25)(10,18,22,37)(11,24,27,33)(13,29,34,26)(14,30,15,32)(17,35,36,38)(19,23,31,39)(20,28,21,40);(3,8)(5,7)(9,13,14,33,16,34,15,24)(10,19,20,35,22,31,21,38)(11,25,26,32,27,12,29,30)(17,18,23,28,36,37,39,40);(2,6)(5,7)

Polycyclic presentation:

2^9|a^2=fg,c^2=i,d^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,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, 6914423);