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×#44589 |
| 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 | |
4834 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);(3,4,8,6)(5,10,7,9)(12,15)(16,18)(19,20,24,28)(21,25,29,33)(22,26,31,34)(23,30,32,27);(3,5,8,7)(4,9,6,10)(21,29)(23,32)(25,33)(27,30);(11,12)(13,16)(14,15)(17,18)(19,21,24,29)(20,25,28,33)(22,30,31,27)(23,26,32,34);(3,6,8,4)(5,9,7,10)(19,22)(20,26)(21,27)(23,25)(24,31)(28,34)(29,30)(32,33);(3,7,8,5)(4,9,6,10)(11,13,14,17)(12,16,15,18)(19,23)(20,27)(21,26)(22,25)(24,32)(28,30)(29,34)(31,33)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7576268);