Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#44856 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4)(5,6)(7,9)(8,10)(13,17)(15,18)(19,20,24,22)(21,25,27,29)(23,30,31,26)(28,34,33,32);(3,5,8,7)(4,9,10,6)(11,12,14,16)(13,15,17,18)(21,27)(23,31)(25,29)(26,30);(11,13)(12,15)(14,17)(16,18)(19,21,24,27)(20,25,22,29)(23,28,31,33)(26,32,30,34);(3,6,8,9)(4,5,10,7)(19,22,24,20)(21,25,27,29)(23,30,31,26)(28,32,33,34);(3,7,8,5)(4,9,10,6)(19,23)(20,26)(21,28)(22,30)(24,31)(25,32)(27,33)(29,34)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7576460);