Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#59598 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2430 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6624 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);(3,4,7,8)(5,6,9,10)(12,16)(13,18)(19,20,23,22)(21,25,26,24);(4,8)(5,9)(12,16)(13,18)(19,21)(20,24)(22,25)(23,26);(11,12)(13,17)(14,18)(15,16)(19,22,23,20)(21,25,26,24);(3,5)(4,6)(7,9)(8,10)(11,13,15,18)(12,17,16,14);(3,6)(4,5)(7,10)(8,9)(11,14,15,17)(12,18,16,13)(19,21)(20,24)(22,25)(23,26)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7573192);