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×#48569 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2407 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5874 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);(3,4)(5,7)(6,10)(8,9)(20,21)(23,31)(25,27)(29,34);(3,5)(4,7)(6,8)(9,10)(11,12)(13,17)(14,18)(15,16)(19,20,24,21)(22,27,30,25)(23,26,31,28)(29,33,34,32);(4,9)(5,6)(12,16)(13,18)(19,21,24,20)(22,27,30,25)(23,28,31,26)(29,33,34,32);(3,6,8,5)(4,10,9,7)(11,13)(12,14)(15,18)(16,17)(19,22,24,30)(20,25,21,27)(23,29,31,34)(26,32,28,33);(3,7,8,10)(4,5,9,6)(11,14)(12,13)(15,17)(16,18)(19,23)(20,26)(21,28)(22,29)(24,31)(25,32)(27,33)(30,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7611007);