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×#39809 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5246 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);(3,4,6,8)(5,7,9,10)(13,17)(15,18)(19,20)(21,23)(22,30)(24,25)(26,29)(27,32)(28,33)(31,34);(11,12,14,16)(13,18,17,15)(20,25)(23,26)(27,30)(31,33);(3,4)(5,7)(6,8)(9,10)(11,13,14,17)(12,15,16,18)(19,21)(20,26)(22,28)(23,25)(24,29)(27,33)(30,31)(32,34);(19,22,24,32)(20,27,25,30)(21,28,29,34)(23,31,26,33);(3,5,6,9)(4,7,8,10)(11,12)(13,15)(14,16)(17,18)(19,23,24,26)(20,21,25,29)(22,31,32,33)(27,28,30,34)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7544555);