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:
(2,6)(7,8)(12,20)(16,21)(18,24)(22,23)(25,26)(27,35)(28,33)(29,31)(30,40)(32,36)(34,37)(38,39);(1,2,5,6)(3,7,4,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27)(26,32)(28,30)(29,35)(31,36)(33,34)(37,38)(39,40);(1,3,5,4)(2,7,6,8)(10,14)(15,17)(16,21)(22,23)(26,29)(27,36)(28,38)(34,40);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,28)(26,33)(27,37)(29,39)(30,36)(31,38)(32,40)(34,35);(1,3,5,4)(2,7,6,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,29,31,26)(27,35,36,32)(28,39,38,33)(30,40,37,34);(1,4,5,3)(2,8,6,7)(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,30,31,37)(26,34,29,40)(27,28,36,38)(32,33,35,39)Polycyclic presentation:
2^9|b^2=gh,c^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7620129);