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 | C4×#3564 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×#538 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6570 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 36 points:
(1,2,3,4);(5,6,10,14)(7,15,16,12)(8,19,22,13)(9,11,21,25)(17,27,29,28)(18,26,30,23)(20,33,34,24)(31,36,32,35);(6,11)(7,16)(8,20)(12,26)(14,25)(15,23)(17,31)(18,30)(22,34)(27,28)(29,32)(35,36);(5,7,9,18)(6,12,11,23)(8,17,20,32)(10,16,21,30)(13,27,24,35)(14,15,25,26)(19,28,33,36)(22,29,34,31);(5,8,21,34)(6,13,25,33)(7,17,30,31)(9,20,10,22)(11,24,14,19)(12,27,26,36)(15,28,23,35)(16,29,18,32);(6,11)(12,23)(13,24)(14,25)(15,26)(19,33)(27,35)(28,36)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533374);