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×#37709 |
| 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 | |
7156 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)(17,23)(18,22)(19,20)(21,25)(24,29)(26,27)(28,30);(4,6)(7,9,11,8)(10,13,14,12)(16,20)(21,23)(22,27)(28,29);(3,4)(5,6)(7,10,11,14)(8,12,9,13)(15,16)(17,21)(18,22)(19,20)(23,25)(24,28)(26,27)(29,30);(15,17,19,25)(16,21,20,23)(18,24,26,30)(22,28,27,29);(3,4)(5,6)(15,18,19,26)(16,22,20,27)(17,24,25,30)(21,28,23,29)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7542310);