#6671658

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

2062 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)(11,17)(12,25)(13,19)(14,29)(15,21)(16,22)(18,26)(20,32)(23,33)(24,27)(28,37)(30,40)(31,35)(34,39)(36,38);(1,2,4,5)(3,6,7,8)(9,11)(10,17)(12,22)(13,23)(14,30)(15,18)(16,25)(19,33)(20,36)(21,26)(24,28)(27,37)(29,40)(31,39)(32,38)(34,35);(9,12,16,17)(10,18,22,23)(11,21,25,19)(13,26,15,33)(14,27,32,34)(20,30,29,38)(24,37,39,35)(28,36,31,40);(1,3,4,7)(2,6,5,8)(9,13,16,15)(10,19,22,21)(11,23,25,18)(12,26,17,33)(14,28,32,31)(20,35,29,37)(24,38,39,30)(27,36,34,40);(2,5)(6,8)(9,14)(10,20)(11,24)(12,27)(13,28)(15,31)(16,32)(17,34)(18,30)(19,35)(21,37)(22,29)(23,38)(25,39)(26,36)(33,40)

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=fi,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, 6578854);