Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C16 · Schur multiplier (H2) = C210 |
| direct product | C22×D4×C16 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C16 | C22 |
| commutator | C2 | C24×C16 |
| Frattini, cosocle | C2×C8 | C25 |
| socle | C24 | C22×C8 |
| prime radical | C23×D4 | C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
2430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2);(3,4);(5,6,7,8);(6,8);(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)320 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C22×#12062, C2×D4×#346, C22×D4×D16, … |
| maximal normal subgroup of | C22×#12062, C2×D4×#346, C22×D4×D16, … |
| maximal quotient of | C22×#12062, #72320682 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529762);