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×#38150 |
| 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 | |
4948 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);(12,15)(16,18)(20,22)(21,27)(26,30)(29,33);(4,7)(5,9)(19,20,24,22)(21,28,27,25)(23,26,31,30)(29,34,33,32);(3,4,6,7)(5,8,9,10)(11,12)(13,16)(14,15)(17,18)(19,21,24,27)(20,25,22,28)(23,29,31,33)(26,32,30,34);(11,13,14,17)(12,16,15,18)(19,22,24,20)(21,28,27,25)(23,30,31,26)(29,34,33,32);(3,5)(4,8)(6,9)(7,10)(19,23,24,31)(20,26,22,30)(21,29,27,33)(25,32,28,34)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7542855);