Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19328 |
| 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 | |
292 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 292 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) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11)(4,10)(5,7)(6,9)(8,12)(13,16)(14,15)(17,18,22,24)(19,21,26,31)(20,27,30,25)(23,29,32,28);(2,7)(8,15)(9,10)(11,16)(17,19,21,24,22,26,31,18)(20,25,29,32,30,27,28,23);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,11)(10,16)(17,20)(18,23)(19,25)(21,29)(22,30)(24,32)(26,27)(28,31);(1,4,5,6)(2,9,7,10)(3,12,13,14)(8,11,15,16)(17,21,22,31)(18,19,24,26)(20,28,30,29)(23,27,32,25)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,d^a=dh,e^a=eg,f^a=fh,d^c=dg,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #300426, #300427, #300434, … |
| maximal normal subgroup of | #300426, #300427, #300434, … |
| maximal quotient of | #189607, #197847, #197850, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12353);