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×#24461 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4545 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)(9,18)(10,24)(11,27)(12,16)(13,17)(14,21)(15,25)(19,23)(20,29)(22,30)(26,31)(28,32)(33,35)(34,37)(36,38);(7,9,13,23)(8,14,17,30)(10,20,26,36)(11,22,12,21)(15,29,32,38)(16,19,27,18)(24,33,31,34)(25,37,28,35);(4,6)(7,8)(9,19)(10,15)(11,16)(12,27)(13,17)(14,21)(18,23)(20,34)(22,30)(24,25)(26,32)(28,31)(29,37)(33,36)(35,38);(7,10,13,26)(8,15,17,32)(9,20,23,36)(11,28,12,25)(14,29,30,38)(16,31,27,24)(18,33,19,34)(21,37,22,35)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345097);