Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C4 | C43 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C23 | C4×D4 |
| prime radical | C24 | #41 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
443 conjugacy classes of subgroups; only the 231 normal subgroups are shown.
This group has 443 conjugacy classes of subgroups — too many to draw. Only its 231 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)(3,4)(5,6,7,8)(9,10)(11,16)(12,15)(13,17)(14,18)(19,24)(20,22)(21,23);(2,4)(10,15,18,17)(11,19)(16,23,24,22)(20,21);(9,11,13,21,14,19,12,20)(10,16,17,23,18,24,15,22)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=g,d^2=h,g^2=h,b^a=bd,d^a=dh,f^a=fh,c^b=ch88 conjugacy classes — view the full character table.
| maximal subgroup of | #128836, #128837, #146326, … |
| maximal normal subgroup of | #128836, #128837, #146326, … |
| maximal quotient of | #105278, #105280, #105281, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3695);