Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | D8×#121 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2,3,4,5,6,7,8);(2,8)(3,7)(4,6);(10,12)(14,16);(13,14,15,16);(9,10)(11,12)98 conjugacy classes — view the full character table.
| maximal subgroup of | #72239637, #72256956, #72257022, … |
| maximal normal subgroup of | #72239637, #72256956, #72257022, … |
| maximal quotient of | #72239676, #72249744, #72264710, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481035);