Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4423 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,13,16)(11,14,17,18)(12,15,21,20)(19,22,23,24)(25,26)(27,33)(28,36)(29,32)(30,35)(31,38)(34,39)(37,40);(1,2)(3,4)(5,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,22)(20,21)(23,24)(25,26,29,32)(27,30,35,33)(28,31,38,36)(34,39,37,40);(2,6)(3,7)(11,17)(14,18)(19,23)(22,24)(25,27,29,35)(26,30,32,33)(28,34,38,37)(31,39,36,40);(3,7)(4,8)(9,10)(11,18)(12,20)(13,16)(14,17)(15,21)(19,22)(23,24)(25,27)(26,30)(28,37)(29,35)(31,40)(32,33)(34,38)(36,39);(1,3)(2,4)(5,7)(6,8)(9,11,13,17)(10,14,16,18)(12,19,21,23)(15,22,20,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,40)(36,38)(37,39);(1,4)(2,3)(5,8)(6,7)(9,12,13,21)(10,15,16,20)(11,19,17,23)(14,22,18,24)(25,28,29,38)(26,31,32,36)(27,34,35,37)(30,39,33,40)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,e^d=egi,f^d=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9532655);