Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32929 |
| automorphism | Aut(G) order 12582912 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1023 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,4)(6,8)(10,13)(11,14)(15,16);(9,10,14,16,12,15,11,13);(1,2,3,4);(5,6,7,8)Polycyclic presentation:
2^8|b^2=eh,c^2=f,d^2=g,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh76 conjugacy classes — view the full character table.
| Fitting subgroup of | #33301⋊C3, #72228326, #72228330 |
| maximal subgroup of | #102229, #102256, #102275, … |
| maximal normal subgroup of | #102229, #102256, #102275, … |
| maximal quotient of | #352877, #352881, #352885, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26289);