#9210278

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

4435 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:

(2,4)(6,8)(9,10,14,16)(11,18,22,15)(12,17,23,21)(13,20,24,19)(25,26,31,30)(27,32,35,36)(28,37,38,33)(29,34,39,40);(3,7)(6,8)(9,11,14,22)(10,15,16,18)(12,20,23,19)(13,21,24,17)(25,27)(26,32)(28,29)(30,36)(31,35)(33,40)(34,37)(38,39);(1,2,5,4)(3,6,7,8)(10,16)(12,23)(15,18)(19,20)(27,35)(28,38)(32,36)(33,37);(9,12)(10,17)(11,20)(13,15)(14,23)(16,21)(18,24)(19,22)(25,28,31,38)(26,33,30,37)(27,29,35,39)(32,40,36,34);(1,3,5,7)(2,6,4,8)(9,11)(10,18)(12,19)(13,17)(14,22)(15,16)(20,23)(21,24)(25,29)(26,34)(27,28)(30,40)(31,39)(32,37)(33,36)(35,38);(1,4,5,2)(3,8,7,6)(9,13)(10,19)(11,21)(12,15)(14,24)(16,20)(17,22)(18,23)(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=g,c^2=gh,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, 9117474);