Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#50894 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2540 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7650 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);(4,6)(8,10)(12,13)(15,16,20,23)(17,21,24,29)(18,19,22,28)(25,30,27,26);(7,8,11,10)(9,13,14,12)(17,24)(18,22)(19,28)(21,29);(3,4,5,6)(15,17)(16,21)(18,27)(19,26)(20,24)(22,25)(23,29)(28,30);(7,9,11,14)(8,12,10,13)(15,18,20,22)(16,19,23,28)(17,25,24,27)(21,30,29,26);(7,10,11,8)(9,13,14,12)(15,19)(16,22)(17,26)(18,23)(20,28)(21,25)(24,30)(27,29)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613916);