Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18618 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
442 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 442 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10)(3,11,14,15)(4,8,5,9)(7,13,16,12)(18,23)(19,27)(20,21)(25,26)(28,29)(30,31);(1,3)(2,7)(4,12)(5,13)(6,14)(8,15)(9,11)(10,16)(17,18,22,26)(19,24,28,32)(20,23,21,25)(27,30,29,31);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,16,11)(17,19,22,28)(18,24,26,32)(20,27,21,29)(23,30,25,31);(1,4,6,5)(2,9,10,8)(3,13,14,12)(7,15,16,11)(17,20,22,21)(18,25,26,23)(19,27,28,29)(24,31,32,30)Polycyclic presentation:
2^8|a^2=gh,b^2=h,c^2=g,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #245390, #245402, #245440, … |
| maximal normal subgroup of | #245390, #245402, #245440, … |
| maximal quotient of | #166460, #166467, #166476, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11643);