Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9369 |
| automorphism | Aut(G) = C23×D4×#120 (order 2048) · Inn(G) = #386 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | #3217 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
249 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 249 conjugacy classes of subgroups — too many to draw. Only its 61 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:
(1,2,4,6,5,3,8,7)(9,10)(11,17)(12,20,22,15)(13,19)(14,16)(18,21,24,23);(1,3)(2,5)(4,6)(7,8)(9,11,17,10,14,19,13,16)(12,18,21,15,22,24,23,20);(1,2)(3,5)(4,6)(7,8)(9,12)(10,15)(11,18)(13,23)(14,22)(16,20)(17,21)(19,24)Polycyclic presentation:
2^8|a^2=f,b^2=dfgh,d^2=gh,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,d^b=dh,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fg37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9378, #200241, #200243, … |
| maximal normal subgroup of | C2×#9378, #200241, #200243, … |
| maximal quotient of | #100049, #100050, #100057, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2366);