Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
365 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 365 conjugacy classes of subgroups — too many to draw. Only its 163 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) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,18,15,19,13,28)(11,16,22,30,23,31,21,37)(12,24,27,32,25,38,26,17)(20,33,36,39,34,40,35,29);(2,5)(6,8)(9,11,14,22,15,23,13,21)(10,16,18,30,19,31,28,37)(12,20,27,36,25,34,26,35)(17,29,24,33,32,39,38,40);(9,12,13,26,15,25,14,27)(10,17,18,24,19,32,28,38)(11,20,21,35,23,34,22,36)(16,29,30,33,31,39,37,40);(1,3,4,7)(2,6,5,8)(12,25)(17,32)(20,34)(24,38)(26,27)(29,39)(33,40)(35,36)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=f,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #124540, #495000, #499478, … |
| maximal normal subgroup of | #124540, #495000, #499478, … |
| maximal quotient of | #206508, #206509, #206510, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14284);