Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33359 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C2×#33372 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C2×Q8 | C25 |
| prime radical | C2×#3499 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×D4 ◁ C22×2−1+4 ◁ G | |
| LECS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
4108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,8,13,9,14,5,15)(6,11,20,27,18,25,16,29)(7,12,17,24,22,28,21,10)(19,26,30,33,31,34,32,23);(3,5,9,8)(4,10,14,24)(6,16,18,20)(7,21,22,17)(11,23,25,33)(12,13,28,15)(19,32,31,30)(26,27,34,29);(3,6)(4,11)(5,16)(7,19)(8,20)(9,18)(10,23)(12,26)(13,27)(14,25)(15,29)(17,30)(21,32)(22,31)(24,33)(28,34);(6,18)(11,25)(16,20)(19,31)(23,33)(26,34)(27,29)(30,32)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529675);