Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C22 | C22×C42 |
| Frattini, cosocle | C24 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
1205 conjugacy classes of subgroups; only the 559 normal subgroups are shown.
This group has 1205 conjugacy classes of subgroups — too many to draw. Only its 559 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) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,13,15,10)(8,11)(9,12)(14,19,20,17)(16,18);(1,2,3,4)(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19);(5,8)(6,11)(7,14)(9,16)(10,17)(12,18)(13,19)(15,20);(8,16)(11,18)(14,20)(17,19)Polycyclic presentation:
2^8|a^2=f,b^2=g,b^a=be,e^a=eh,f^b=fh,d^c=dh82 conjugacy classes — view the full character table.
| maximal subgroup of | #138232, #138276, #138278, … |
| maximal normal subgroup of | #138232, #138276, #138278, … |
| maximal quotient of | #180999, #181011, #181012, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13342);