Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C28×C4 |
| direct product | C4×#1779 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C42 | C23 |
| commutator | C22 | C22×C4×C8 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C24 | C23×C4 |
| prime radical | C26 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
6990 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4);(5,6)(7,8)(9,10)(11,12)(13,14,15,17,16,18,19,20);(6,8);(10,12)224 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#32878, #72286322, #72286329, … |
| maximal normal subgroup of | C4×#32878, #72286322, #72286329, … |
| maximal quotient of | #72286655 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261971);