#7143638

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993899
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

2149 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)(9,10,15,19)(11,21,25,40)(12,26,23,36)(13,18,31,37)(14,32,22,20)(16,33,35,24)(17,27,34,28)(29,39,30,38);(2,5)(3,7)(9,11,22,33,15,25,14,24)(10,16,32,40,19,35,20,21)(12,13,28,30,23,31,27,29)(17,18,26,39,34,37,36,38);(1,2)(3,6)(4,5)(7,8)(9,12,15,23)(10,17,19,34)(11,13,25,31)(14,27,22,28)(16,18,35,37)(20,36,32,26)(21,38,40,39)(24,29,33,30);(2,5)(6,8)(9,13)(10,18)(11,23)(12,25)(14,30)(15,31)(16,34)(17,35)(19,37)(20,39)(21,36)(22,29)(24,27)(26,40)(28,33)(32,38);(1,3,4,7)(2,6,5,8)(10,19)(16,35)(17,34)(18,37)(20,32)(21,40)(26,36)(38,39)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,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, 7050834);