#8700360

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

4514 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8607556);