Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C23×C4 |
| direct product | M16×M32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C4×C8 | C24 |
| commutator, socle | C22 | C22×C4×C8 |
| prime radical | C24 | C4×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
645 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,4,5,6,7,8);(2,6)(4,8);(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24);(10,18)(12,20)(14,22)(16,24)200 conjugacy classes — view the full character table.
| maximal subgroup of | M32×#131, #72310100, M32×AGL(1,ℤ/8), … |
| maximal normal subgroup of | M32×#131, #72310100, M32×AGL(1,ℤ/8), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387115);