#8417098

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

4230 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:

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

Polycyclic presentation:

2^9|a^2=i,b^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, 8324294);