Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C29 |
| direct product | C2×#8299 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×C22⋊C4 |
| commutator | C23 | C23×C8 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C26 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
5574 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)(7,8)(9,13,14,11)(10,12)(15,16,17,19,18,20,21,22);(7,9,10,14)(8,11,12,13);(4,6)128 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#184774 |
| maximal normal subgroup of | C2×#184774 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 91932);