Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #15257 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C22×#121 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1175 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(10,12)(13,14)(15,18)(16,17)(19,20);(6,8)(10,12)(14,16)(18,19);(5,6)(7,8)(9,10,11,12)(13,15,17,20)(14,18,16,19);(1,2)(3,4)(13,16,17,14)(15,19,20,18)Polycyclic presentation:
2^8|c^2=ef,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#15319, #608852, #609015, … |
| maximal normal subgroup of | C2×#15319, #608852, #609015, … |
| maximal quotient of | #154690, #154729, #154761, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8307);