Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
| direct product | C2×#34372 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6810 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 38 points:
(1,2);(3,4,5,6)(7,8,12,10)(9,13,17,15)(11,18,19,14)(16,21,22,20)(24,28)(25,31)(26,34)(36,38);(8,10)(13,15)(14,18)(20,21)(23,24,27,28)(25,32,31,29)(26,30,34,35)(33,38,37,36);(7,9,12,17)(8,13,10,15)(11,16,19,22)(14,20,18,21)(23,24)(25,29)(26,30)(27,28)(31,32)(33,36)(34,35)(37,38);(7,10,12,8)(9,15,17,13)(11,18,19,14)(16,21,22,20)(23,25,27,31)(24,29,28,32)(26,33,34,37)(30,36,35,38);(7,11)(8,14)(9,16)(10,18)(12,19)(13,20)(15,21)(17,22)(23,26)(24,30)(25,33)(27,34)(28,35)(29,36)(31,37)(32,38)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533171);