Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#24462 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#578 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4497 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(3,4,5,6)(7,8,13,17)(9,19,22,29)(10,23,27,15)(11,21,14,30)(12,18,31,16)(20,36,38,33)(24,32,37,25)(26,34,28,35);(7,9,13,22)(8,14,17,11)(10,24,27,37)(12,30,31,21)(15,33,23,36)(16,29,18,19)(20,26,38,28)(25,34,32,35);(4,6)(7,10,13,27)(8,15,17,23)(9,20,22,38)(11,25,14,32)(12,26,31,28)(16,34,18,35)(19,33,29,36)(21,37,30,24);(7,11)(8,9)(10,25)(12,19)(13,14)(15,20)(16,30)(17,22)(18,21)(23,38)(24,34)(26,33)(27,32)(28,36)(29,31)(35,37)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345096);