Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | #3050 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
292 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 111 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)(11,15)(12,16)(13,26)(14,17)(18,23)(19,24)(20,27)(21,36)(22,28)(25,31)(29,33)(30,34)(32,37)(35,38)(39,40);(2,5)(6,8)(9,11,14,22)(10,15,17,28)(12,20,25,35)(13,21,24,34)(16,27,31,38)(18,29,32,39)(19,30,26,36)(23,33,37,40);(1,3,4,7)(2,6,5,8)(10,16)(12,23)(13,24)(15,27)(17,31)(18,19)(20,33)(21,34)(25,37)(26,32)(28,38)(29,30)(35,40)(36,39);(10,17)(15,28)(16,31)(18,32)(19,26)(27,38)(29,39)(30,36)Polycyclic presentation:
2^8|b^2=h,c^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fg,g^a=gh,f^c=fg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #495950, #495952, #496885, … |
| maximal normal subgroup of | #495950, #495952, #496885, … |
| maximal quotient of | #493901, #494239, #494444, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26635);