#6682653

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682648
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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 ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2156 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,h^b=hi,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, 6589849);