#8754836

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

4322 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)(6,8)(9,10)(11,17)(12,15)(13,16)(14,20)(18,22)(19,23)(21,24)(26,31)(27,35)(28,38)(34,39);(3,7)(6,8)(9,11,15,13)(10,16,12,17)(14,21,22,19)(18,24,20,23)(25,26,30,31)(27,36,35,32)(28,33,38,40)(29,39,37,34);(1,2)(3,6)(4,5)(7,8)(10,12)(16,17)(18,20)(23,24)(25,27,30,35)(26,32,31,36)(28,37,38,29)(33,39,40,34);(9,12,15,10)(11,17,13,16)(14,20,22,18)(19,24,21,23)(25,28)(26,33)(27,37)(29,35)(30,38)(31,40)(32,39)(34,36);(1,3)(2,6)(4,7)(5,8)(9,13,15,11)(10,17,12,16)(14,21,22,19)(18,24,20,23)(25,29,30,37)(26,34,31,39)(27,28,35,38)(32,33,36,40);(1,2)(3,6)(4,5)(7,8)(9,14)(10,18)(11,19)(12,20)(13,21)(15,22)(16,23)(17,24)(25,28)(26,33)(27,37)(29,35)(30,38)(31,40)(32,39)(34,36)

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662032);