Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1047 conjugacy classes of subgroups; only the 195 normal subgroups are shown.
This group has 1047 conjugacy classes of subgroups — too many to draw. Only its 195 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10)(11,20)(12,17)(13,18)(14,19)(15,22)(16,21)(23,24);(2,4)(5,6)(7,8)(9,11)(10,15)(12,21)(13,22)(14,23)(16,19)(17,20)(18,24);(6,8)(9,11)(10,16)(12,21)(13,22)(14,23)(15,19)(17,24)(18,20);(9,12,14,13)(10,17,19,18)(11,21,23,22)(15,20,16,24)Polycyclic presentation:
2^8|d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=cg,e^b=eh,f^b=fh,e^c=eh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #99711, #99714, #150182, … |
| maximal normal subgroup of | #99711, #99714, #150182, … |
| maximal quotient of | #232199, #232206, #232207, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16409);