Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#41393 |
| 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 | |
4994 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);(4,6)(5,9)(13,17)(15,18)(19,20,24,25)(21,23,27,33)(22,29,32,26)(28,34,30,31);(3,4,7,6)(5,10,9,8)(11,12,14,16)(13,15,17,18)(20,25)(23,33)(26,29)(31,34);(3,5,7,9)(4,8,6,10)(11,13)(12,15)(14,17)(16,18)(19,21,24,27)(20,23,25,33)(22,30,32,28)(26,34,29,31);(3,6,7,4)(5,10,9,8)(19,22,24,32)(20,26,25,29)(21,28,27,30)(23,31,33,34);(19,23)(20,27)(21,25)(22,31)(24,33)(26,30)(28,29)(32,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7546982);