Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C28×C4 |
| direct product | C4×#1786 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C24 | C22×D4 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
5938 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)(16,19)(18,20);(14,17)(15,20)(18,19);(6,8)152 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#21452, #72286292, #72286304, … |
| maximal normal subgroup of | C4×#21452, #72286292, #72286304, … |
| maximal quotient of | #72279520, #72286524, C4×#12736, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 262001);