Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61643 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61647 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7806 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)(13,20)(14,15)(17,24)(18,22)(21,23)(25,26);(3,5)(4,7)(6,10)(8,9)(11,12)(13,17)(14,22)(15,18)(16,19)(20,24)(21,26)(23,25);(11,13)(12,17)(14,20)(15,23)(16,21)(18,25)(19,26)(22,24);(5,10)(7,9)(11,14,16,15)(12,18,19,22)(13,20,21,23)(17,25,26,24);(4,8)(7,9)(12,19)(17,26)(18,22)(24,25)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10484446);