Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9495 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#9500 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C26 | C23 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
1894 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4)(5,8,12,15)(6,9)(7,10)(11,14,17,18)(13,16)(19,20)(21,24)(22,29,30,25)(23,26)(27,31,33,34)(28,32);(3,5)(4,8,10,15)(6,11)(7,12)(9,14,16,18)(13,17)(19,21,23,28)(20,24,26,32)(22,27,30,33)(25,31,29,34);(3,6,7,13)(4,9,10,16)(5,11,12,17)(8,14,15,18)(19,22)(20,25)(21,27)(23,30)(24,31)(26,29)(28,33)(32,34)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 108848);