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) = C24×C4 |
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 | C2×#121 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
583 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 583 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)(13,14,18,20,16,21,17,22)(15,23,26,27,24,19,25,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,24)(17,25)(18,26)(20,28)(21,23)(22,27);(2,4)(6,9)(10,12)(14,20)(15,24)(17,18)(19,27)(21,22)(23,28);(5,7,8,11)(6,10,9,12)(14,21)(19,23)(20,22)(27,28)Polycyclic presentation:
2^8|a^2=g,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=ce,f^c=fh,g^c=gh64 conjugacy classes — view the full character table.
| maximal subgroup of | #107369, #108007, #119260, … |
| maximal normal subgroup of | #107369, #108007, #119260, … |
| maximal quotient of | #216050, #216054, #216083, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14849);