Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #117 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C2 | C2×#432 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ #405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
264 conjugacy classes of subgroups; only the 157 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 157 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,10,5,11,17,28,6,9,18,24,19,25,31,32)(3,12,14,26,15,22,30,27,13,7,16,20,29,8,21,23);(1,3)(2,7)(4,16)(5,15)(6,13)(8,11)(9,12)(10,26)(14,18)(17,21)(19,29)(20,24)(22,25)(23,32)(27,28)(30,31);(1,3)(2,8,9,22)(4,16)(5,15)(6,13)(7,11,12,25)(10,27,24,23)(14,18)(17,21)(19,29)(20,32,26,28)(30,31);(3,13)(7,12)(8,22)(14,16)(15,29)(20,26)(21,30)(23,27)Polycyclic presentation:
2^8|a^2=f,c^2=e,f^2=g,g^2=h,b^a=be,c^a=cg,e^a=eh,d^b=dh,f^b=fh,d^c=dh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #126722, #126723, #126872, … |
| maximal normal subgroup of | #126722, #126723, #126872, … |
| maximal quotient of | #492926, #492956, #493112, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26416);