Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #102 |
| commutator | C8 | C22×C8 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C22 | C2×#102 |
| prime radical | C2×#135 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
197 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 197 conjugacy classes of subgroups — too many to draw. Only its 79 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:
(1,2,3,5,4,6,7,8)(9,10)(11,20,23,37)(12,16)(13,27)(14,19)(15,22,31,39)(17,25)(18,26)(21,36,24,33)(28,35,40,30)(29,32)(34,38);(9,11)(10,15)(12,21)(13,28)(14,23)(16,30)(17,33)(18,35)(19,31)(20,32)(22,38)(24,26)(25,39)(27,37)(29,40)(34,36);(9,12,14,26)(10,16,19,18)(11,21,23,24)(13,25,29,38)(15,30,31,35)(17,32,34,27)(20,36,37,33)(22,28,39,40)Polycyclic presentation:
2^8|a^2=e,c^2=h,d^2=f,e^2=g,f^2=h,b^a=bd,d^a=df,f^a=fh,d^b=dfh,e^b=eh,f^b=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #128537, #128592, #128605, … |
| maximal normal subgroup of | #128537, #128592, #128605, … |
| maximal quotient of | #122464, #122470, #122476, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4585);