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×#39173 |
| 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 | |
7178 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)(17,20)(18,22)(19,27)(21,24)(23,29)(25,30)(26,28);(7,8,10,12)(9,13,14,11)(15,17,21,18)(16,22,24,20)(19,25,29,26)(23,30,27,28);(3,4,5,6)(7,8)(9,11)(10,12)(13,14)(15,18,21,17)(16,22,24,20)(19,26,29,25)(23,30,27,28);(15,19,21,29)(16,23,24,27)(17,25,18,26)(20,28,22,30);(7,9,10,14)(8,11,12,13)(15,20,21,22)(16,17,24,18)(19,28,29,30)(23,25,27,26)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7544325);