Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19384 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
379 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 379 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,6)(7,8)(9,10)(11,15)(12,17)(13,22)(14,18)(16,19)(20,24)(21,23);(1,2,6,8,4,7,3,5)(9,11)(10,15)(12,21)(13,19)(14,20)(16,22)(17,23)(18,24);(9,11)(10,16)(12,19)(13,21)(14,20)(15,22)(17,24)(18,23);(9,12,14,13)(10,17,18,22)(11,19,20,21)(15,16,24,23)Polycyclic presentation:
2^8|b^2=eg,d^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^b=dh,f^c=fh49 conjugacy classes — view the full character table.
| maximal subgroup of | #271899, #271900, #271901, … |
| maximal normal subgroup of | #271899, #271900, #271901, … |
| maximal quotient of | #189139, #197864, #197941, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12398);