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×#60830 |
| 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 | |
7966 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)(11,12)(13,21)(14,20)(15,17)(16,25)(18,22)(19,23)(24,26);(3,5,6,10)(4,7,8,9)(12,17)(18,26)(19,21)(20,25);(5,10)(7,9)(11,13)(12,18)(14,22)(15,23)(16,24)(17,26)(19,20)(21,25);(4,8)(7,9)(11,13)(12,19)(14,24)(15,23)(16,22)(17,21)(18,20)(25,26);(11,14,15,16)(12,20,17,25)(13,22,23,24)(18,19,26,21)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482733);