#9210294

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

4351 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)(3,5)(4,8)(6,7)(9,10,15,14)(11,18,13,16)(12,17,20,21)(19,24,22,23)(25,26,31,30)(27,32,35,36)(28,37,38,33)(29,34,39,40);(2,7)(3,8)(9,11,15,13)(10,16,14,18)(12,19,20,22)(17,23,21,24)(25,27)(26,32)(28,29)(30,36)(31,35)(33,40)(34,37)(38,39);(1,3,6,8)(2,4,7,5)(12,20)(17,21)(19,22)(23,24)(27,35)(28,38)(32,36)(33,37);(1,4,6,5)(2,3,7,8)(9,12)(10,17)(11,19)(13,22)(14,21)(15,20)(16,23)(18,24)(25,28,31,38)(26,33,30,37)(27,29,35,39)(32,40,36,34);(1,2)(3,4)(5,8)(6,7)(9,13,15,11)(10,16,14,18)(12,19,20,22)(17,24,21,23)(25,29)(26,34)(27,28)(30,40)(31,39)(32,37)(33,36)(35,38);(1,5,6,4)(2,8,7,3)(9,14,15,10)(11,18,13,16)(12,21,20,17)(19,24,22,23)(25,30,31,26)(27,36,35,32)(28,37,38,33)(29,34,39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9117490);