#6684987

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682658
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

1918 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fg,d^2=fi,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,d^c=di,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, 6592183);