Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#36494 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2414 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6310 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);(3,4)(5,8)(6,7)(9,10)(20,23)(24,26);(3,4)(5,9)(6,7)(8,10)(11,12)(13,17)(14,18)(15,16)(19,20,22,23)(21,24,25,26);(4,7)(8,9)(12,16)(13,18);(3,4)(5,9)(6,7)(8,10)(11,13,15,18)(12,17,16,14)(19,21,22,25)(20,24,23,26);(3,5,6,10)(4,8,7,9)(11,14,15,17)(12,13,16,18)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7537376);