Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24283 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
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 | |
451 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 451 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 — 4 generators on 20 points:
(6,10)(7,14)(8,12)(13,18)(15,16)(17,19);(1,2,3,4)(5,6,8,10,9,13,12,18)(7,11,15,17,16,20,14,19);(2,4)(5,7)(6,11)(8,15)(9,16)(10,17)(12,14)(13,20)(18,19);(5,6,12,18,9,13,8,10)(7,11,14,19,16,20,15,17)Polycyclic presentation:
2^8|b^2=eg,d^2=eh,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,d^b=dh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #332327, #332332, #332337, … |
| maximal normal subgroup of | #332327, #332332, #332337, … |
| maximal quotient of | #317968, #317972, #317976, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21231);