#7143618

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

2146 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)(9,10,16,19)(11,15,20,14)(12,23,26,21)(13,18,30,27)(17,32,34,25)(22,29,36,31)(24,38,28,37)(33,39,35,40);(2,5)(3,7)(9,11,14,10,16,20,15,19)(12,23,25,34,26,21,32,17)(13,27,29,36,30,18,31,22)(24,33,39,37,28,35,40,38);(1,2)(3,6)(4,5)(7,8)(9,12)(10,17)(11,21)(13,28)(14,25)(15,32)(16,26)(18,35)(19,34)(20,23)(22,38)(24,30)(27,33)(29,40)(31,39)(36,37);(2,5)(6,8)(9,13)(10,18)(11,22)(12,24)(14,31)(15,29)(16,30)(17,33)(19,27)(20,36)(21,37)(23,38)(25,40)(26,28)(32,39)(34,35);(1,3,4,7)(2,6,5,8)(10,19)(11,20)(17,34)(18,27)(21,23)(22,36)(33,35)(37,38)

Polycyclic presentation:

2^9|a^2=i,b^2=f,e^2=hi,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, 7050814);