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×#51490 |
| 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 | |
4984 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)(5,9)(6,8)(7,10)(11,12,14,16)(13,15,17,18)(19,20)(21,26)(22,23)(24,25)(27,32)(28,33)(29,34)(30,31);(3,4,6,8)(5,7,10,9)(20,25)(21,28)(23,32)(29,31);(11,12)(13,15)(14,16)(17,18)(19,21,24,28)(20,26,25,33)(22,31,27,29)(23,30,32,34);(3,4)(5,7)(6,8)(9,10)(19,22,24,27)(20,23,25,32)(21,29,28,31)(26,34,33,30);(3,5)(4,7)(6,10)(8,9)(11,13,14,17)(12,15,16,18)(19,23)(20,27)(21,30)(22,25)(24,32)(26,29)(28,34)(31,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614509);