Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | Aut(G) = C26×#442 (order 4096) · Inn(G) = C22×D8 · Out(G) = C24×C4 |
| 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 | #466 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
248 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,18)(11,20,24,35)(12,17,15,27)(14,30,29,37)(16,31,21,25)(19,28,33,22)(23,34,26,40)(32,38,36,39);(2,5)(6,8)(9,10)(11,21)(12,27)(13,18)(14,19)(15,17)(16,24)(20,31)(22,30)(23,32)(25,35)(26,36)(28,37)(29,33)(34,39)(38,40);(9,11,22,38,15,26,29,31,13,24,28,39,12,23,14,25)(10,16,30,34,17,32,33,35,18,21,37,40,27,36,19,20);(1,3,4,7)(2,6,5,8)(9,12,13,15)(10,17,18,27)(11,23,24,26)(14,28,29,22)(16,32,21,36)(19,30,33,37)(20,34,35,40)(25,39,31,38)Polycyclic presentation:
2^8|a^2=h,c^2=fg,d^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,c^b=ch,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#33745, #7623240, #7623250, … |
| maximal normal subgroup of | C2×#33745, #7623240, #7623250, … |
| maximal quotient of | #494208, #495304, #495305, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26733);