Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| 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 | C2×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
274 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 274 conjugacy classes of subgroups — too many to draw. Only its 103 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)(9,10,15,20)(11,16,24,32)(12,18,14,28)(13,17,27,34)(19,25,33,26)(21,30,23,38)(22,29,37,40)(31,35,39,36);(2,5)(6,8)(9,11,15,24)(10,16,20,32)(12,21,14,23)(13,22,27,37)(17,29,34,40)(18,30,28,38)(19,31,33,39)(25,35,26,36);(10,17)(12,14)(13,26)(16,29)(18,19)(20,34)(21,23)(22,36)(25,27)(28,33)(30,31)(32,40)(35,37)(38,39);(1,3,4,7)(2,6,5,8)(9,12,15,14)(10,18,20,28)(11,21,24,23)(13,25,27,26)(16,30,32,38)(17,33,34,19)(22,35,37,36)(29,39,40,31)Polycyclic presentation:
2^8|a^2=h,b^2=h,d^2=eh,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 | #137025, #137026, #137033, … |
| maximal normal subgroup of | #137025, #137026, #137033, … |
| maximal quotient of | #494034, #494967, #494969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26664);