Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2196 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#12765 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| socle | C22 | #1790 |
| prime radical | C2×#3481 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2761 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);(3,4,6,10,7,11,15,18)(5,9,13,17,12,8,14,16);(4,8)(5,12)(9,11)(10,17)(13,14)(16,18);(5,12)(13,14)80 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#511819, #72231758, #72232800, … |
| maximal normal subgroup of | C2×#511819, #72231758, #72232800, … |
| maximal quotient of | #72231002, #72231276, #72231744, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400784);