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×#37705 |
| 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 | |
7200 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);(7,8,11,9)(10,13,14,12)(15,16)(17,21)(18,25)(19,20)(22,26)(23,29)(24,27)(28,30);(4,6)(7,9,11,8)(10,13,14,12)(16,20)(21,27)(22,25)(28,29);(3,4)(5,6)(7,10,11,14)(8,12,9,13)(15,17,19,24)(16,21,20,27)(18,23,26,30)(22,28,25,29);(15,18,19,26)(16,22,20,25)(17,23,24,30)(21,28,27,29);(3,4)(5,6)(15,16)(17,21)(18,22)(19,20)(23,28)(24,27)(25,26)(29,30)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7542240);