#8347388

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4604 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8254584);