#8418233

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

4273 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)(3,4)(5,6)(7,8)(9,10)(11,16)(12,22)(13,17)(14,18)(15,19)(20,24)(21,23)(27,28)(29,37)(32,34)(33,38);(2,6)(3,7)(9,11)(10,16)(12,14)(13,23)(15,20)(17,21)(18,22)(19,24)(25,26,31,30)(27,34,28,32)(29,38,37,33)(35,39,36,40);(1,3,5,7)(2,4,6,8)(11,20)(14,21)(16,24)(18,23)(25,27,31,28)(26,32,30,34)(29,35,37,36)(33,39,38,40);(1,4)(2,7)(3,6)(5,8)(9,12)(10,13)(11,21)(14,20)(15,17)(16,18)(19,22)(23,24)(25,28,31,27)(26,32,30,34)(29,35,37,36)(33,40,38,39);(9,13)(10,17)(11,18)(12,19)(14,16)(15,22)(20,23)(21,24)(25,29,31,37)(26,33,30,38)(27,35,28,36)(32,39,34,40);(9,14,15,21)(10,18,19,23)(11,12,20,17)(13,16,22,24)(25,30,31,26)(27,32,28,34)(29,38,37,33)(35,39,36,40)

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

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, 8325429);