Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #385 |
| prime radical | C24 | C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
210 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 75 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10,13,17,14,18,24,30)(11,15,21,27,20,26,34,38)(12,16,23,29,22,28,35,39)(19,31,33,40,32,25,36,37);(2,5)(3,8)(6,7)(9,11,13,21,14,20,24,34)(10,15,17,27,18,26,30,38)(12,19,23,33,22,32,35,36)(16,25,29,37,28,31,39,40);(9,12)(10,16)(11,19)(13,23)(14,22)(15,25)(17,29)(18,28)(20,32)(21,33)(24,35)(26,31)(27,37)(30,39)(34,36)(38,40)Polycyclic presentation:
2^8|a^2=f,b^2=f,d^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,d^b=dg,e^b=eh,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #135974, #135975, #135980, … |
| maximal normal subgroup of | #135974, #135975, #135980, … |
| maximal quotient of | #99313, #99316, #99319, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1849);