Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| direct product | D42×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×D42 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
11325 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 16 points:
(1,2,3,4);(2,4);(5,6,7,8);(6,8);(9,10,11,12)(13,16,15,14);(9,13,11,15)(10,14,12,16)125 conjugacy classes — view the full character table.
| maximal subgroup of | D42×#41, D4×#3373, #72281274, … |
| maximal normal subgroup of | D42×#41, D4×#3373, #72281274, … |
| maximal quotient of | #72257697, #72263970, D4×#2409, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7606662);