Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #386 |
| prime radical | #392 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
306 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 306 conjugacy classes of subgroups — too many to draw. Only its 75 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:
(1,2,3,6,4,7,8,5)(9,10)(11,18,22,24)(12,21,23,15)(13,16)(14,17)(19,20);(2,5)(3,8)(6,7)(9,11,19,21,14,22,13,15)(10,12,20,18,17,23,16,24);(9,12)(10,15)(11,20)(13,24)(14,23)(16,22)(17,21)(18,19)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=g,f^2=gh,b^a=bd,c^a=ce,e^a=eh,d^b=dg,e^b=eh,f^b=fg,d^c=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #131298, #131299, #131312, … |
| maximal normal subgroup of | #131298, #131299, #131312, … |
| maximal quotient of | #99551, #99553, #99559, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2023);