Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #127 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #392 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
395 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 395 conjugacy classes of subgroups — too many to draw. Only its 127 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 — 3 generators on 20 points:
(1,2,3,4)(5,6)(7,14)(8,19)(9,12)(10,13)(11,16)(15,17)(18,20);(5,7,8,16,10,17,9,18)(6,11,12,14,13,20,19,15);(5,6)(7,15,17,14)(8,19)(9,12)(10,13)(11,16,20,18)Polycyclic presentation:
2^8|a^2=g,b^2=d,c^2=e,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #93964, #136139, #136140, … |
| maximal normal subgroup of | #93964, #136139, #136140, … |
| maximal quotient of | #121335, #121336, #121337, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4323);