Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
| direct product | C4×#1504 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2052 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4);(5,6)(7,11,12,9)(8,10)(14,17,18,19);(5,7)(6,9)(8,12)(10,11)(13,14)(15,19)(16,18)(17,20);(7,12)(9,11)128 conjugacy classes — view the full character table.
| maximal subgroup of | #72239122, #72239136, #72239137, … |
| maximal normal subgroup of | #72239122, #72239136, #72239137, … |
| maximal quotient of | AGL(1,𝔽5)×#1504 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 89808);