Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18866 |
| automorphism | Aut(G) order 8192 |
| 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 | |
427 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 427 conjugacy classes of subgroups — too many to draw. Only its 121 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)(17,18,23,28)(19,29,25,32)(20,30,24,31)(21,26,22,27);(1,3)(2,7)(4,12)(5,13)(6,14)(8,15)(9,11)(10,16)(17,19,23,25)(18,24,28,20)(21,31,22,30)(26,29,27,32);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,16,11)(17,20,23,24)(18,25,28,19)(21,29,22,32)(26,31,27,30);(1,4,6,5)(2,9,10,8)(3,13,14,12)(7,15,16,11)(17,21,23,22)(18,26,28,27)(19,30,25,31)(20,29,24,32)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=g,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dgh,e^a=eg,f^a=fh,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fh43 conjugacy classes — view the full character table.
| maximal subgroup of | #245430, #245432, #245434, … |
| maximal normal subgroup of | #245430, #245432, #245434, … |
| maximal quotient of | #167186, #167196, #167206, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11875);