Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| direct product | D4×#548 |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
11679 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);(6,8)(10,13)(14,15);(9,10,12,13)(11,15,16,14);(5,6)(7,8);(9,11,12,16)(10,14,13,15)125 conjugacy classes — view the full character table.
| maximal subgroup of | D42×#41, #72281033, #72283295, … |
| maximal normal subgroup of | D42×#41, #72281033, #72283295, … |
| maximal quotient of | #72257495, #72257549, #72257597, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7606664);