Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3186 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
403 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 403 conjugacy classes of subgroups — too many to draw. Only its 199 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:
(1,2,3,4)(6,9)(7,11)(13,14)(15,22)(16,24)(17,21)(18,19)(20,25)(23,28)(26,27);(5,6)(7,10)(8,9)(11,12)(13,15,17,18)(14,19,21,22)(16,25,26,23)(20,24,28,27);(6,9)(10,12)(13,16,18,25,17,26,15,23)(14,20,22,24,21,28,19,27);(5,7,8,11)(6,10,9,12)(14,21)(19,22)(20,28)(24,27)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=f,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce64 conjugacy classes — view the full character table.
| maximal subgroup of | #116762, #116763, #116789, … |
| maximal normal subgroup of | #116762, #116763, #116789, … |
| maximal quotient of | #217464, #217473, #217488, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14973);