#8417290

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

4201 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)(3,4)(5,6)(7,8)(9,10,15,14)(11,16,19,22)(12,21,23,17)(13,20,24,18)(25,26,32,35)(27,36,31,33)(28,30,29,34)(37,39,38,40);(2,6)(3,7)(9,11)(10,16)(12,20)(13,17)(14,22)(15,19)(18,23)(21,24)(25,27,32,31)(26,33,35,36)(28,38,29,37)(30,39,34,40);(1,3,5,7)(2,4,6,8)(10,14)(11,19)(17,21)(18,20)(25,28,32,29)(26,34,35,30)(27,37,31,38)(33,39,36,40);(1,4)(2,7)(3,6)(5,8)(9,12)(10,17)(11,20)(13,16)(14,21)(15,23)(18,19)(22,24)(25,29,32,28)(26,30,35,34)(27,38,31,37)(33,40,36,39);(9,13,15,24)(10,18,14,20)(11,21,19,17)(12,22,23,16)(25,30)(26,28)(27,39)(29,35)(31,40)(32,34)(33,38)(36,37);(9,14,15,10)(11,22,19,16)(12,21,23,17)(13,20,24,18)(25,31,32,27)(26,33,35,36)(28,38,29,37)(30,40,34,39)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,d^2=ghi,e^2=gi,f^2=h,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, 8324486);