Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9443 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#9444 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1272 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
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)(21,28,29,32)(22,25,30,31)(23,26)(24,27)(33,34);(3,5)(4,8,10,15)(6,11)(7,12)(9,14,16,18)(13,17)(19,21,23,31,24,29,33,25)(20,22,26,32,27,30,34,28);(3,6,7,13)(4,9,10,16)(5,11,12,17)(8,14,15,18)(19,22)(20,25)(21,26)(23,32)(24,30)(27,31)(28,33)(29,34)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 108274);