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) = C22×C4 |
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 | |
283 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 283 conjugacy classes of subgroups — too many to draw. Only its 67 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:
(1,2,6,7,5,3,4,8)(9,10)(11,19,22,23)(12,16,15,21)(13,17)(14,18)(20,24);(1,3,5,2)(4,8,6,7)(9,11,20,21,14,22,13,16)(10,15,24,19,18,12,17,23);(1,4,5,6)(2,7,3,8)(9,12,14,15)(10,16,18,21)(11,17,22,24)(13,19,20,23)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=g,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 | #132853, #132854, #132855, … |
| maximal normal subgroup of | #132853, #132854, #132855, … |
| maximal quotient of | #99552, #99554, #99560, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2024);