Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8325 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #379 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
342 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 79 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) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,27)(20,31,33,24)(21,34)(22,26)(23,29)(25,36)(28,42,44,41)(30,40)(32,39,48,46)(35,38)(37,43,45,47);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,22,30)(18,23,26,38)(20,28,33,44)(21,35,36,29)(24,37,31,45)(25,40,34,27)(32,47,48,43)(39,42,46,41);(3,9)(6,12)(10,15)(13,16)(17,20,22,33)(18,24,26,31)(19,28,30,44)(21,32,36,48)(23,37,38,45)(25,39,34,46)(27,41,40,42)(29,43,35,47)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=g,d^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=ch,d^b=dg,f^c=fg46 conjugacy classes — view the full character table.
| maximal subgroup of | #132095, #132097, #132098, … |
| maximal normal subgroup of | #132095, #132097, #132098, … |
| maximal quotient of | #96992, #96994, #96997, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1315);