#9206643

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

4164 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gh,d^2=gh,e^2=ghi,f^2=h,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, 9113839);