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×#49715 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4768 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);(11,12,15,18)(13,16,14,17)(19,20)(21,25)(22,26)(23,24)(27,29)(28,33)(30,34)(31,32);(4,7)(5,9)(11,13,15,14)(12,16,18,17)(20,23)(21,27)(26,31)(28,30);(3,4,6,7)(5,8,9,10)(11,12)(13,16)(14,17)(15,18)(19,21,24,27)(20,25,23,29)(22,30,32,28)(26,34,31,33);(3,4)(5,8)(6,7)(9,10)(11,14,15,13)(12,17,18,16)(19,22,24,32)(20,26,23,31)(21,28,27,30)(25,33,29,34);(3,5)(4,8)(6,9)(7,10)(19,23,24,20)(21,29,27,25)(22,31,32,26)(28,34,30,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7612691);