Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#46139 |
| 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 | |
4952 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)(5,6)(7,8)(9,10)(13,17)(15,18)(19,20,23,26)(21,27,29,24)(22,25,31,30)(28,33,34,32);(4,7)(5,9)(11,12,14,16)(13,15,17,18)(19,20,23,26)(21,27,29,24)(22,30,31,25)(28,32,34,33);(3,5)(4,6)(7,10)(8,9)(11,13)(12,15)(14,17)(16,18)(19,20)(21,24)(22,25)(23,26)(27,29)(28,32)(30,31)(33,34);(3,6,8,10)(4,9,7,5)(19,21,23,29)(20,24,26,27)(22,28,31,34)(25,32,30,33);(3,7,8,4)(5,10,9,6)(19,22)(20,25)(21,28)(23,31)(24,32)(26,30)(27,33)(29,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7609980);