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 | |
4310 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,14)(12,16)(13,17)(15,18)(19,21)(20,23)(22,24)(25,26)(27,33)(28,34)(29,32)(30,31)(35,40)(36,37)(38,39);(1,2,5,6)(3,7,4,8)(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,24)(22,23)(25,27)(26,28)(29,37)(30,34)(31,33)(32,40)(35,38)(36,39);(1,3,5,4)(2,7,6,8)(11,18)(12,20)(14,15)(16,23)(26,31)(27,34)(32,39)(35,37);(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,39)(38,40);(1,3,5,4)(2,7,6,8)(9,12,13,20)(10,16,17,23)(11,19,18,24)(14,21,15,22)(25,27)(26,28)(29,35)(30,34)(31,33)(32,36)(37,38)(39,40);(1,4,5,3)(2,8,6,7)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,29,30,38)(26,32,31,39)(27,35,34,37)(28,36,33,40)Polycyclic presentation:
2^9|b^2=gh,c^2=gh,e^2=gi,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, 7649784);