Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×D4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | C2×#95 |
| prime radical | C24 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
318 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 318 conjugacy classes of subgroups — too many to draw. Only its 131 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)(3,5)(4,7)(6,8)(9,10,12,16,14,18,13,17)(11,15,20,23,19,22,21,24);(2,5,7,6)(9,11,13,21,14,19,12,20)(10,15,17,24,18,22,16,23);(11,19)(15,22)(20,21)(23,24)Polycyclic presentation:
2^8|a^2=e,b^2=f,d^2=g,e^2=h,f^2=h,b^a=bd,d^a=dg,f^a=fg,c^b=ch76 conjugacy classes — view the full character table.
| maximal subgroup of | #105772, #105773, #106180, … |
| maximal normal subgroup of | #105772, #105773, #106180, … |
| maximal quotient of | #94885, #94886, #94900, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 871);