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) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
231 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 231 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) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(6,7)(9,10,14,11)(12,15,23,18)(13,20,22,17)(16,24,19,21);(5,6)(7,8)(9,11,14,10)(12,18,23,15)(13,17,22,20)(16,21,19,24);(2,4)(5,7,8,6)(10,15)(11,18)(12,21)(13,22)(16,17)(19,20)(23,24)Polycyclic presentation:
2^8|a^2=h,b^2=h,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 | #125662, #125663, #125664, … |
| maximal normal subgroup of | #125662, #125663, #125664, … |
| maximal quotient of | #144888, #144933, #145008, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6397);