Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#46992 |
| 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 | |
4822 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);(4,6)(8,9)(11,12,15,18)(13,16,14,17)(19,20)(21,26)(22,27)(23,30)(24,25)(28,33)(29,32)(31,34);(3,4,7,6)(5,9,10,8)(11,13,15,14)(12,16,18,17)(20,25)(21,29)(22,31)(23,33);(3,4,7,6)(5,8,10,9)(11,12)(13,16)(14,17)(15,18)(19,21,24,29)(20,26,25,32)(22,28,31,30)(23,27,33,34);(3,5,7,10)(4,8,6,9)(19,22,24,31)(20,27,25,34)(21,30,29,28)(23,32,33,26);(3,6,7,4)(5,9,10,8)(11,14,15,13)(12,17,18,16)(19,23)(20,28)(21,27)(22,32)(24,33)(25,30)(26,31)(29,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7607905);