Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13357 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C23 |
| prime radical | C23 | #120 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
192 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 192 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,5,7)(3,8,4,6)(10,13)(11,15)(14,20)(17,19)(18,21)(23,24);(1,3,5,4)(2,6,7,8)(9,10,12,13)(11,17,18,24)(14,22,20,16)(15,23,21,19);(1,4,5,3)(2,6,7,8)(9,11,12,18)(10,15,13,21)(14,23,20,19)(16,24,22,17)Polycyclic presentation:
2^8|a^2=dh,b^2=d,c^2=d,e^2=h,f^2=h,b^a=bd,c^a=ce,e^a=eh,f^a=fgh,c^b=cf,e^b=eg,f^b=fh,e^c=eh,f^c=fh,g^c=gh,f^e=fh31 conjugacy classes — view the full character table.
| maximal subgroup of | #151406, #151407, #151410, … |
| maximal normal subgroup of | #151406, #151407, #151410, … |
| maximal quotient of | #145778, #145779, #145794, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6453);