#7143615

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993875
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

2122 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,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, 7050811);