#7007243

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996799
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

1959 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)(11,23,26,39)(12,28,25,40)(13,32)(14,21)(15,22)(16,20)(17,27,36,30)(18,31,38,33)(19,34)(24,35)(29,37);(1,2)(3,7)(4,6)(5,8)(9,11)(10,17)(12,29)(13,33)(14,30)(15,27)(16,26)(18,37)(19,28)(20,36)(21,39)(22,23)(24,25)(31,34)(32,40)(35,38);(1,3)(2,5)(4,8)(6,7)(9,12,16,25)(10,18,20,38)(11,24,26,29)(13,27,34,30)(14,31,15,33)(17,35,36,37)(19,39,32,23)(21,28,22,40);(3,8)(5,7)(9,13,14,24,16,34,15,29)(10,19,21,37,20,32,22,35)(11,25,27,33,26,12,30,31)(17,18,23,28,36,38,39,40);(2,6)(5,7)(10,20)(11,26)(13,34)(18,38)(21,22)(24,29)(27,30)(28,40)

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,e^d=ei

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, 6914439);