Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24378 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#27975 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,7,10)(5,9,6,8)(11,12,14,16)(13,15,17,18)(19,20,23,25)(21,29,30,34)(22,28,24,27)(26,31,33,32);(3,5,7,6)(4,8,10,9)(11,13)(12,15)(14,17)(16,18)(20,25)(21,30)(27,28)(31,32);(3,5,7,6)(4,9,10,8)(13,17)(15,18)(19,21,22,31,23,30,24,32)(20,26,27,34,25,33,28,29);(3,6,7,5)(4,9,10,8)(19,22,23,24)(20,27,25,28)(21,31,30,32)(26,34,33,29)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6365684);