#8757526

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

4493 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)(3,7)(4,8)(5,6)(9,10)(11,17)(12,15)(13,16)(14,20)(18,22)(19,23)(21,24)(25,26,31,33)(27,32,37,40)(28,34,36,30)(29,39,38,35);(1,2,5,6)(3,7,8,4)(9,11,15,13)(10,16,12,17)(14,21,22,19)(18,24,20,23)(25,27)(26,32)(28,38)(29,36)(30,39)(31,37)(33,40)(34,35);(1,3)(2,4)(5,8)(6,7)(10,12)(16,17)(18,20)(23,24)(26,33)(27,37)(28,36)(35,39);(1,4)(2,3)(5,7)(6,8)(9,12,15,10)(11,17,13,16)(14,20,22,18)(19,24,21,23)(25,28,31,36)(26,34,33,30)(27,38,37,29)(32,35,40,39);(1,2)(3,4)(5,6)(7,8)(9,13,15,11)(10,17,12,16)(14,21,22,19)(18,24,20,23)(25,29,31,38)(26,35,33,39)(27,36,37,28)(30,32,34,40);(1,4)(2,3)(5,7)(6,8)(9,14)(10,18)(11,19)(12,20)(13,21)(15,22)(16,23)(17,24)(25,30,31,34)(26,36,33,28)(27,35,37,39)(29,32,38,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=gi,e^2=gi,f^2=gh,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, 8664722);