Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C82 · Schur multiplier (H2) = C23×C8 |
| direct product | C8×D4⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C8 | D4 |
| commutator | C4 | C2×C82 |
| Frattini, cosocle | C43 | C23 |
| socle | C23 | C4×C22⋊C4 |
| prime radical | C22×D4 | C42 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,5,4,6,7,8);(9,10)(11,13)(12,15)(14,16)(17,18,19,21,20,22,23,24);(10,13)(11,16)(14,15)224 conjugacy classes — view the full character table.
| maximal subgroup of | C8×#1580, C2×C8×D4⋊C8 |
| maximal normal subgroup of | C8×#1580, C2×C8×D4⋊C8 |
| maximal quotient of | C8×#1333, C2×C8×D4⋊C8, C8×#1336 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3880);