#8720723

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

4342 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=hi,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8627919);