Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#41939 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7172 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)(15,16,21,23)(17,24,18,20)(19,27,29,22)(25,28,26,30);(7,8)(9,10)(11,12)(13,14)(15,17,21,18)(16,20,23,24)(19,25,29,26)(22,28,27,30);(3,4)(5,6)(8,12)(9,13)(15,18,21,17)(16,20,23,24)(19,25,29,26)(22,30,27,28);(7,9,11,13)(8,10,12,14)(15,19,21,29)(16,22,23,27)(17,25,18,26)(20,28,24,30);(7,10)(8,13)(9,12)(11,14)(15,20)(16,17)(18,23)(19,28)(21,24)(22,25)(26,27)(29,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7547610);