Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1685 |
| automorphism | Aut(G) = C25×D42 (order 2048) · Inn(G) = #121 · Out(G) = C26 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #386 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
228 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 228 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:
(2,5)(3,7)(6,8)(9,10)(11,18,20,14)(12,17)(13,16)(15,19,24,22)(21,23);(1,2,3,5,4,6,7,8)(9,11,12,20)(10,14)(13,22,21,19)(15,23)(16,24)(17,18);(2,6)(5,8)(9,11)(10,15)(12,20)(13,22)(14,16)(17,24)(18,23)(19,21)Polycyclic presentation:
2^8|a^2=f,b^2=df,d^2=gh,e^2=h,b^a=bd,c^a=ce,d^a=dg,c^b=cg,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=eh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9209, #197577, #197578, … |
| maximal normal subgroup of | C2×#9209, #197577, #197578, … |
| maximal quotient of | #99859, #99860, #99863, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2197);