Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62295 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#62583 |
| 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 | |
7364 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);(3,4,8,10)(5,6,9,7)(11,12)(13,20)(14,18)(15,24)(16,19)(17,22)(21,26)(23,25);(11,13)(12,17)(14,21)(15,23)(16,22)(18,25)(19,20)(24,26);(3,5,7,4,8,9,6,10)(11,14,16,15)(12,18,19,24)(13,21,22,23)(17,25,20,26);(13,22)(17,20)(21,23)(25,26);(3,6,8,7)(4,9,10,5)(11,15,16,14)(12,18,19,24)(13,23,22,21)(17,25,20,26)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10487168);