Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#53254 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2688 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5184 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);(4,7)(5,6)(13,17)(16,18)(19,20,23,26)(21,24,28,31)(22,25,29,32)(27,30,33,34);(3,4,8,7)(5,10,6,9)(12,15)(16,18)(22,29)(25,32)(27,33)(30,34);(3,5,8,6)(4,9,7,10)(11,12)(13,16)(14,15)(17,18)(19,20)(21,24)(22,25)(23,26)(27,30)(28,31)(29,32)(33,34);(3,6,8,5)(4,10,7,9)(11,13)(12,16)(14,17)(15,18)(19,21,23,28)(20,24,26,31)(22,27,29,33)(25,30,32,34);(3,7,8,4)(5,10,6,9)(11,12)(13,16)(14,15)(17,18)(19,22)(20,25)(21,27)(23,29)(24,30)(26,32)(28,33)(31,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012905);