Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19186 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #121 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
281 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 281 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2)(3,10)(4,9)(5,7)(6,13)(8,11)(12,16)(14,15)(17,18,22,26)(19,27,31,41)(20,24,32,38)(21,34,28,25)(23,35,37,30)(29,40,43,48)(33,45,42,39)(36,46,47,44);(2,7)(8,15)(9,13)(10,16)(17,19,28,35,22,31,21,30)(18,23,34,41,26,37,25,27)(20,29,42,46,32,43,33,44)(24,36,45,48,38,47,39,40);(1,3)(2,8)(4,11)(5,12)(6,14)(7,15)(9,16)(10,13)(17,20)(18,24)(19,29)(21,33)(22,32)(23,36)(25,39)(26,38)(27,40)(28,42)(30,44)(31,43)(34,45)(35,46)(37,47)(41,48);(1,4,5,6)(2,9,7,13)(3,11,12,14)(8,16,15,10)(20,32)(24,38)(29,43)(33,42)(36,47)(39,45)(40,48)(44,46)Polycyclic presentation:
2^8|a^2=g,b^2=egh,d^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^c=dg,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #138554, #138565, #489328, … |
| maximal normal subgroup of | #138554, #138565, #489328, … |
| maximal quotient of | #189583, #197774, #202770, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12214);