Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60771 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212×C4 |
| direct product | C2×#60825 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7950 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 26 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(12,16)(14,21)(17,22)(18,23)(20,25)(24,26);(3,5,6,10)(4,7,8,9)(12,17)(16,22)(18,24)(23,26);(5,10)(7,9)(11,12)(13,18)(14,22)(15,17)(16,21)(19,24)(20,26)(23,25);(4,8)(7,9)(11,13)(12,18)(14,20)(15,19)(16,23)(17,24)(21,25)(22,26);(13,19)(18,24)(20,25)(23,26)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482728);