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×#48686 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2407 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5896 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)(13,14)(15,17)(16,18)(19,20,24,22)(21,26,29,25)(23,30,32,27)(28,33,31,34);(3,4)(5,9)(6,10)(7,8)(13,15)(14,17)(19,21,24,29)(20,25,22,26)(23,31,32,28)(27,34,30,33);(4,8)(5,10)(11,13,16,15)(12,17,18,14)(19,21)(20,26)(22,25)(23,28)(24,29)(27,34)(30,33)(31,32);(3,5)(4,6)(7,10)(8,9)(11,14,16,17)(12,13,18,15)(19,22,24,20)(21,26,29,25)(23,30,32,27)(28,34,31,33);(3,6)(4,5)(7,9)(8,10)(11,15,16,13)(12,17,18,14)(19,23,24,32)(20,27,22,30)(21,28,29,31)(25,33,26,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7611222);