Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #95 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | C2×#95 |
| prime radical | C25 | Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
453 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 453 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,16,14,18,13,17)(11,19,20,15,22,23,21,24);(1,3)(2,5)(4,8)(6,7)(9,11,13,21,14,22,12,20)(10,15,16,23,18,24,17,19);(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,23,24,19)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=gh,d^2=gh,e^2=gh,f^2=gh,b^a=bd,d^a=dg,f^a=fgh,e^b=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #93036, #93048, #105325, … |
| maximal normal subgroup of | #93036, #93048, #105325, … |
| maximal quotient of | #95069, #95070, #95078, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 996);