#8464222

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

4471 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=ghi,d^2=h,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8371418);