Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#62044 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3546 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6666 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);(4,7)(5,9)(11,12,14,17)(13,19,20,26)(15,23,22,18)(16,24,25,21);(3,4)(5,8)(6,7)(9,10)(11,12)(13,16)(14,17)(15,18)(19,21)(20,25)(22,23)(24,26);(11,13,14,20)(12,16,17,25)(15,24,22,21)(18,26,23,19);(4,7)(8,10)(12,17)(16,25)(18,23)(19,26);(3,5,6,9)(4,8,7,10)(12,17)(16,25)(18,23)(19,26)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10486032);