Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C23 |
| commutator | C22 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C23×C4 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
887 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10,13,17,14,12,16,21)(11,15,20,23,19,18,22,24);(2,4)(5,6,7,8)(9,11)(10,15)(12,18)(13,20)(14,19)(16,22)(17,23)(21,24);(9,12,13,21,14,10,16,17)(11,18,20,24,19,15,22,23);(11,19)(15,18)(20,22)(23,24)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=f,f^2=h,b^a=be,d^b=dh112 conjugacy classes — view the full character table.
| maximal subgroup of | #105325, #105326, #105418, … |
| maximal normal subgroup of | #105325, #105326, #105418, … |
| maximal quotient of | #181887, #181894, #181900, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13373);