Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#37771 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2656 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7894 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);(7,8,10,12)(9,11,14,13)(15,16,20,21)(17,24,22,18)(19,23,27,28)(25,30,29,26);(4,6)(7,8,10,12)(9,13,14,11)(16,21)(17,22)(23,28)(25,29);(3,4)(5,6)(7,8)(9,11)(10,12)(13,14)(15,17,20,22)(16,18,21,24)(19,25,27,29)(23,26,28,30);(15,18)(16,22)(17,21)(19,26)(20,24)(23,29)(25,28)(27,30);(3,4)(5,6)(7,9,10,14)(8,11,12,13)(15,19,20,27)(16,23,21,28)(17,25,22,29)(18,26,24,30)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7542190);