Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#35625 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2407 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8336 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 30 points:
(1,2);(3,4,5,6)(7,8,12,14)(9,11,13,10)(16,20)(21,23)(22,27)(28,29);(7,9,12,13)(8,10,14,11)(16,20)(18,26)(21,23)(24,30);(3,4)(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,23)(18,22)(19,20)(21,25)(24,29)(26,27)(28,30);(7,10,12,11)(8,9,14,13)(15,17,19,25)(16,21,20,23)(18,24,26,30)(22,28,27,29);(7,11,12,10)(8,13,14,9)(15,18)(16,22)(17,24)(19,26)(20,27)(21,28)(23,29)(25,30)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536224);