Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#40120 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5004 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(11,12,14,16)(13,15,18,17)(19,20)(21,26)(22,30)(23,28)(24,25)(27,32)(29,31)(33,34);(4,8)(6,9)(11,12,14,16)(13,17,18,15)(20,25)(23,26)(27,30)(31,33);(3,4)(5,9)(6,10)(7,8)(11,12)(13,15)(14,16)(17,18)(19,21,24,28)(20,26,25,23)(22,29,32,34)(27,33,30,31);(3,5,7,10)(4,9,8,6)(19,22)(20,27)(21,29)(23,31)(24,32)(25,30)(26,33)(28,34);(3,6,7,9)(4,10,8,5)(11,13,14,18)(12,15,16,17)(19,23,24,26)(20,28,25,21)(22,31,32,33)(27,34,30,29)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7544976);