Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#51555 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C26 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5248 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4)(5,8)(6,10)(7,9)(13,17)(15,18);(3,5)(4,6)(7,10)(8,9)(11,12,14,16)(13,15,17,18)(20,24)(21,27)(22,30)(32,34);(3,5,7,10)(4,6,9,8)(11,13)(12,15)(14,17)(16,18)(19,20)(21,28)(22,26)(23,24)(25,27)(29,34)(30,31)(32,33);(3,5)(4,8)(6,9)(7,10)(19,21)(20,25)(22,29)(23,27)(24,28)(26,32)(30,33)(31,34);(3,6,7,8)(4,5,9,10)(19,22,23,30)(20,26,24,31)(21,29,27,33)(25,32,28,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614711);