#8726533

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

4491 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8633729);