#9706325

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

4472 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=h,d^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9613521);