Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | Aut(G) = C26×#442 (order 4096) · Inn(G) = C22×D8 · Out(G) = C26 |
| 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 | #2186 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
277 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 277 conjugacy classes of subgroups — too many to draw. Only its 101 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 — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,21)(12,18)(13,19)(14,17)(15,29)(16,23)(20,28)(22,36)(24,35)(25,31)(26,33)(27,34)(30,39)(32,38)(37,40);(2,5)(6,8)(9,11,13,23)(10,16,19,21)(12,22,15,25)(14,24,27,39)(17,30,34,35)(18,31,29,36)(20,32,33,37)(26,38,28,40);(10,17)(12,15)(14,28)(16,30)(18,20)(19,34)(21,35)(22,25)(24,40)(26,27)(29,33)(31,32)(36,37)(38,39);(1,3,4,7)(2,6,5,8)(9,12,13,15)(10,18,19,29)(11,22,23,25)(14,26,27,28)(16,31,21,36)(17,33,34,20)(24,38,39,40)(30,37,35,32)Polycyclic presentation:
2^8|b^2=h,d^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,d^c=dh,f^c=fg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #483877, #483890, #483923, … |
| maximal normal subgroup of | #483877, #483890, #483923, … |
| maximal quotient of | #494033, #494967, #494968, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26659);