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×#3568 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×#3568 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6676 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);(5,6,10,12)(7,16,20,18)(8,11,15,21)(9,23,24,14)(13,27,26,17)(19,32,33,30)(22,34,29,25)(28,31,35,36);(6,12)(7,17)(8,15)(13,18)(14,23)(16,26)(19,31)(20,27)(22,29)(28,32)(30,35)(33,36);(5,7,10,20)(6,13,12,26)(8,18,15,16)(9,19,24,33)(11,17,21,27)(14,28,23,35)(22,32,29,30)(25,31,34,36);(5,8,21,6)(7,18,27,13)(9,22,34,14)(10,15,11,12)(16,17,26,20)(19,32,36,28)(23,24,29,25)(30,31,35,33);(5,9,21,34)(6,14,8,22)(7,19,27,36)(10,24,11,25)(12,23,15,29)(13,28,18,32)(16,30,26,35)(17,31,20,33)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533376);