Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| socle | C22 | C22×D8 |
| prime radical | #2215 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
263 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 263 conjugacy classes of subgroups — too many to draw. Only its 107 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)(11,16)(12,22)(13,27)(14,29)(15,19)(17,24)(18,23)(20,33)(21,32)(25,39)(26,36)(28,40)(30,34)(31,38)(35,37);(2,5)(6,8)(9,11,15,14)(10,16,19,29)(12,20,26,25)(13,21,28,23)(17,30,31,35)(18,27,32,40)(22,33,36,39)(24,34,38,37);(9,12,23,37,14,25,28,38,15,26,21,34,11,20,13,24)(10,17,27,33,16,30,32,36,19,31,40,39,29,35,18,22);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|b^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=gh76 conjugacy classes — view the full character table.
| maximal subgroup of | #150119, #150120, #150123, … |
| maximal normal subgroup of | #150119, #150120, #150123, … |
| maximal quotient of | #494197, #495295, #495296, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26723);