Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32944 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
516 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 516 conjugacy classes of subgroups — too many to draw. Only its 147 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) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(6,10)(7,13)(8,16)(11,17)(14,15)(18,20);(5,6,8,10,9,11,16,17)(7,12,14,18,15,19,13,20);(1,2,3,4)(6,11)(10,17)(12,19)(18,20);(5,7)(6,12)(8,14)(9,15)(10,18)(11,19)(13,16)(17,20)Polycyclic presentation:
2^8|b^2=e,c^2=f,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gh,c^b=ch,g^d=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #273691, #273692, #273699, … |
| maximal normal subgroup of | #273691, #273692, #273699, … |
| maximal quotient of | #351770, #351771, #351776, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26181);