Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12711 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#12711 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | C2×#3484 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
4112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4)(5,12,14,8)(6,10)(7,11)(9,13,17,15)(16,18);(4,8)(9,10)(11,12)(17,18);(3,5)(4,9)(6,15)(7,14)(8,10)(11,17)(12,18)(13,16)68 conjugacy classes — view the full character table.
| maximal subgroup of | #72214353, #72214565, #72215601, … |
| maximal normal subgroup of | #72214353, #72214565, #72215601, … |
| maximal quotient of | #72214353, #72231038, #72231069, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400602);