Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | Aut(G) = C24×#3484 (order 2048) · Inn(G) = #120 · Out(G) = C26 |
| 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 | #3100 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
272 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 69 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:
(1,2,3,7,5,8,4,6)(9,10,13,17,14,11,15,19)(12,18,21,24,20,16,22,23);(2,6)(3,4)(7,8)(9,11,13,19,14,10,15,17)(12,18,22,23,20,16,21,24);(2,7)(3,4)(6,8)(9,12)(10,16)(11,18)(13,22)(14,20)(15,21)(17,24)(19,23)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=g,e^2=g,f^2=h,b^a=bd,c^a=ce,e^a=egh,d^b=dg,e^b=eh,f^b=fg,d^c=dh,e^c=eg,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9049, #195354, #195355, … |
| maximal normal subgroup of | C2×#9049, #195354, #195355, … |
| maximal quotient of | #99581, #99583, #99589, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2037);