Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22⋊C4 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C22 | C2×C22⋊C8 |
| prime radical | C23 | C22⋊C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
163 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 163 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5)(3,7)(6,8)(9,10,12,17,14,18,25,32,13,16,24,29,26,30,37,39)(11,19,21,31,22,33,35,40,20,15,23,27,34,28,36,38);(1,2,4,6)(3,8,7,5)(9,11,13,20)(10,15,16,19)(12,23,24,21)(14,22,26,34)(17,31,29,27)(18,28,30,33)(25,36,37,35)(32,40,39,38);(11,20)(15,19)(21,23)(22,34)(27,31)(28,33)(35,36)(38,40)Polycyclic presentation:
2^8|a^2=e,b^2=f,d^2=fh,e^2=g,g^2=h,b^a=bd,d^a=df,c^b=ch,d^b=dfh,e^b=eh76 conjugacy classes — view the full character table.
| maximal subgroup of | #128680, #128681, #128700, … |
| maximal normal subgroup of | #128680, #128681, #128700, … |
| maximal quotient of | #124076, #124115, #124133, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4598);