Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| 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 | C2×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
276 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 276 conjugacy classes of subgroups — too many to draw. Only its 105 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)(9,10,12,17)(11,20,21,15)(13,16,24,31)(14,27,25,19)(18,33,32,26)(22,34,37,28)(23,30,38,36)(29,39,35,40);(2,5)(6,8)(9,11,12,21)(10,15,17,20)(13,22,24,37)(14,23,25,38)(16,28,31,34)(18,29,32,35)(19,30,27,36)(26,39,33,40);(10,16)(13,26)(14,25)(15,28)(17,31)(18,19)(20,34)(22,39)(23,38)(24,33)(27,32)(29,30)(35,36)(37,40);(1,3,4,7)(2,6,5,8)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,f^c=fg,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #137011, #137012, #137019, … |
| maximal normal subgroup of | #137011, #137012, #137019, … |
| maximal quotient of | #494028, #494960, #494962, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26655);