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×#39266 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5538 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)(19,20,24,23)(21,27,29,25)(22,30,31,26)(28,32,34,33);(11,12,14,16)(13,17,18,15)(20,23)(25,27)(26,30)(32,33);(3,4)(5,7)(6,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,21,24,29)(20,25,23,27)(22,28,31,34)(26,32,30,33);(19,22)(20,26)(21,28)(23,30)(24,31)(25,32)(27,33)(29,34);(3,5,6,9)(4,7,8,10)(11,13,14,18)(12,15,16,17)(19,23,24,20)(21,27,29,25)(22,30,31,26)(28,33,34,32)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7544142);