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 | |
4543 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,14,16)(11,17,18,12)(13,15,22,24)(19,23,21,20)(25,26,31,33)(27,35,38,32)(28,29,39,40)(30,34,37,36);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,21)(14,18)(15,23)(16,17)(19,22)(20,24)(25,27,31,38)(26,32,33,35)(28,36,39,34)(29,37,40,30);(11,18)(12,17)(19,21)(20,23)(26,33)(28,39)(32,35)(34,36);(9,11)(10,12)(13,19)(14,18)(15,20)(16,17)(21,22)(23,24)(25,28)(26,29)(27,36)(30,35)(31,39)(32,37)(33,40)(34,38);(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,24)(22,23)(25,29)(26,28)(27,37)(30,38)(31,40)(32,36)(33,39)(34,35);(1,3,4,7)(2,5,6,8)(9,13,14,22)(10,15,16,24)(11,19,18,21)(12,20,17,23)(25,30,31,37)(26,34,33,36)(27,29,38,40)(28,35,39,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7646721);