Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63011 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C219 |
| direct product | C2×#63039 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
34620 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 7 generators on 26 points:
(1,2);(3,4,8,7)(5,10,6,9)(11,12,16,19)(13,20,23,17)(14,15,18,24)(21,26,25,22);(3,5,8,6)(4,9,7,10)(11,12,16,19)(13,20,23,17)(14,15,18,24)(21,26,25,22);(3,5,8,6)(4,9,7,10)(11,13,16,23)(12,17,19,20)(14,21,18,25)(15,22,24,26);(3,5,8,6)(4,9,7,10);(3,6,8,5)(4,9,7,10)(11,14)(12,15)(13,21)(16,18)(17,22)(19,24)(20,26)(23,25);(3,7,8,4)(5,10,6,9)(11,15)(12,18)(13,22)(14,19)(16,24)(17,25)(20,21)(23,26)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493557);