Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11663 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#12735 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C23×#135 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
3330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(3,4)(5,6)(8,12,13,21)(9,15)(10,18)(14,16,19,22)(17,20);(7,8)(9,16)(10,19)(11,13)(12,20)(14,18)(15,21)(17,22);(4,6)92 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#495721, #72250120, #72250124, … |
| maximal normal subgroup of | C2×#495721, #72250120, #72250124, … |
| maximal quotient of | #72265396, #72280466, C2×#129503, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400717);