#8505392

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

4445 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=i,d^2=i,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=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, 8412588);