Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C4×#2080 |
| subgroup | G/H | |
|---|---|---|
| center | C42 | #124 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#397 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1008 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4);(5,6)(7,15,18,28)(8,12)(9,13)(10,14)(11,17,25,22)(16,31,19,30)(20,26)(21,27)(23,29)(24,33,35,32)(34,36);(5,7)(6,11)(8,19)(9,22)(10,18)(12,24)(13,28)(14,25)(15,29)(16,21)(17,23)(20,32)(26,31)(27,35)(30,36)(33,34);(5,8,9,20,10,21,23,34)(6,12,13,26,14,27,29,36)(7,16,17,32,18,19,22,33)(11,24,15,30,25,35,28,31)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266471);