#8688266

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

4364 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8595462);