Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| 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 | #532 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
361 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 361 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)(9,10,15,20,16,21,14,32)(11,22,26,34,27,38,25,17)(12,28,13,19,30,36,29,18)(23,35,31,40,39,33,24,37);(2,5)(6,8)(9,11,14,25,16,27,15,26)(10,17,20,22,21,34,32,38)(12,23,29,24,30,39,13,31)(18,33,28,37,19,35,36,40);(1,3,4,7)(2,6,5,8)(9,12,14,29,16,30,15,13)(10,18,20,28,21,19,32,36)(11,23,25,24,27,39,26,31)(17,33,22,37,34,35,38,40);(9,13,15,30,16,29,14,12)(10,19,20,36,21,18,32,28)(11,24,26,23,27,31,25,39)(17,33,22,37,34,35,38,40)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=e,d^2=g,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #124769, #219562, #219563, … |
| maximal normal subgroup of | #124769, #219562, #219563, … |
| maximal quotient of | #212684, #212685, #212686, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14647);