Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#493 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
283 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 283 conjugacy classes of subgroups — too many to draw. Only its 69 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) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(6,7)(9,10)(11,14)(12,21)(13,17)(15,18)(16,19)(20,23)(22,24);(5,6)(7,8)(9,10)(11,14)(12,21)(13,17)(15,18)(16,19)(20,23)(22,24);(2,4)(5,7,8,6)(10,14)(11,18)(12,19)(15,16)(17,23)(20,24)(21,22)Polycyclic presentation:
2^8|c^2=d,e^2=gh,f^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=eg,f^a=fg,g^a=gh,c^b=cf,e^b=eg,f^b=fg,g^b=gh,e^c=eg,f^c=fg,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #125658, #125659, #125660, … |
| maximal normal subgroup of | #125658, #125659, #125660, … |
| maximal quotient of | #144886, #144931, #145005, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6395);