Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13343 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | B(4,𝔽2) |
| commutator, Frattini, prime radical, cosocle | C22×D4 | C23 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
242 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 242 conjugacy classes of subgroups — too many to draw. Only its 49 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,17)(12,14)(13,16)(15,19)(18,23)(20,22)(21,24);(1,3,5,4)(2,6,7,8)(10,14)(13,20)(15,21)(17,23);(1,4,5,3)(2,6,7,8)(9,11,12,18)(10,15,14,21)(13,19,20,24)(16,17,22,23)Polycyclic presentation:
2^8|a^2=dh,b^2=dh,c^2=d,b^a=bd,c^a=ce,f^a=fgh,c^b=cf,e^b=eg,g^c=gh,f^e=fh31 conjugacy classes — view the full character table.
| maximal subgroup of | #151402, #151403, #151412, … |
| maximal normal subgroup of | #151402, #151403, #151412, … |
| maximal quotient of | #145776, #145778, #145792, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6451);