#6671652

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667760
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

2080 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,h^c=hi,h^e=hi

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, 6578848);