Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24384 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#532 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
667 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 667 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,11)(8,10)(9,12)(13,14)(15,23)(16,22)(17,19)(18,21)(20,24)(25,28)(26,27);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(14,19)(20,27)(21,22)(23,28);(2,4)(5,7,9,8)(6,11,12,10)(13,15,16,26,17,25,18,24)(14,20,21,28,19,27,22,23);(5,8,9,7)(6,11,12,10)(13,16,17,18)(14,21,19,22)(15,24,25,26)(20,23,27,28)Polycyclic presentation:
2^8|b^2=eh,c^2=ef,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #152771, #152773, #246769, … |
| maximal normal subgroup of | #152771, #152773, #246769, … |
| maximal quotient of | #303119, #303126, #303133, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18850);