#7149413

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993899
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 ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1848 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:

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

Polycyclic presentation:

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

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, 7056609);