Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D82 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | D8×#102 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×D4×D8 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,9)(8,11)(10,12);(6,9)(7,12)(10,11);(1,2,3,4)(13,14)(15,17)(16,19)(18,20);(14,17)(15,20)(18,19)98 conjugacy classes — view the full character table.
| maximal subgroup of | C4×D82, #72226890, #72226892, … |
| maximal normal subgroup of | C4×D82, #72226890, #72226892, … |
| maximal quotient of | #72216412, #72225399, D8×#357, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 127996);