Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#35125 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2464 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5892 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 26 points:
(1,2);(4,7)(5,9)(12,13)(17,18)(20,23)(24,26);(3,4,8,7)(5,10,9,6)(11,12,16,13)(14,17,15,18)(19,20)(21,24)(22,23)(25,26);(3,5)(4,6)(7,10)(8,9)(11,13,16,12)(14,18,15,17);(3,6,8,10)(4,9,7,5)(11,14,16,15)(12,17,13,18);(3,7,8,4)(5,10,9,6)(11,15,16,14)(12,18,13,17)(19,21,22,25)(20,24,23,26)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7535530);