Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#46861 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5034 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);(3,4,8,6)(5,9,10,7)(12,15)(16,18)(20,21)(23,31)(25,27)(29,34);(3,5)(4,9)(6,7)(8,10)(19,20,24,21)(22,27,30,25)(23,26,31,28)(29,33,34,32);(4,6)(5,10)(11,12)(13,16)(14,15)(17,18)(19,21,24,20)(22,27,30,25)(23,28,31,26)(29,33,34,32);(3,6,8,4)(5,7,10,9)(11,13,14,17)(12,16,15,18)(19,22,24,30)(20,25,21,27)(23,29,31,34)(26,32,28,33);(3,7)(4,5)(6,10)(8,9)(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, 7607834);