Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #15257 |
| automorphism | Aut(G) order 131072 |
| 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 | C26 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1417 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 1417 conjugacy classes of subgroups — too many to draw. Only its 187 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6,7,8)(13,14,16,19)(15,18,20,17);(6,8)(10,12)(13,15,16,20)(14,17,19,18);(5,6)(7,8)(9,10)(11,12)(13,15)(14,18)(16,20)(17,19);(1,2)(3,4)(5,6)(7,8)(13,14)(15,18)(16,19)(17,20)Polycyclic presentation:
2^8|a^2=e,b^2=ef,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 | #142570, C2×#15284, #608824, … |
| maximal normal subgroup of | #142570, C2×#15284, #608824, … |
| maximal quotient of | #154676, #154677, #154702, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8272);