Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33752 |
| automorphism | Aut(G) order 8192 |
| 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 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
233 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 233 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) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,29)(20,28)(21,24)(22,35)(23,26)(25,32)(27,37)(30,45)(31,47)(33,40)(34,42)(36,44)(38,43)(39,46)(41,48);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(18,24)(19,30)(26,28)(27,40)(31,38)(32,46)(35,42)(36,48);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,19,21,30)(18,25,24,39)(20,31,23,38)(22,36,34,48)(26,41,28,44)(27,43,40,47)(29,42,45,35)(32,33,46,37);(2,8)(6,9)(7,16)(14,15)(17,20,21,23)(18,26,24,28)(19,31,30,38)(22,33,34,37)(25,41,39,44)(27,42,40,35)(29,43,45,47)(32,36,46,48)Polycyclic presentation:
2^8|b^2=eh,c^2=e,d^2=h,f^2=gh,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fg,g^a=gh,c^b=ce,f^c=fg,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #489403, #489404, #489431, … |
| maximal normal subgroup of | #489403, #489404, #489431, … |
| maximal quotient of | #496431, #496479, #496737, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26787);