Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8516 |
| 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 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
328 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 328 conjugacy classes of subgroups — too many to draw. Only its 71 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:
(1,2,5,10,6,9,4,8)(3,11,12,16,13,7,14,15)(17,18)(19,27,30,39)(20,31,33,24)(21,25)(22,26)(23,29,37,34)(28,42)(32,46,48,38)(35,40)(36,43)(41,44)(45,47);(1,3,6,13)(2,7,9,11)(4,12,5,14)(8,15,10,16)(17,19)(18,23)(20,28)(21,34)(22,30)(24,36)(25,39)(26,37)(27,40)(29,35)(31,45)(32,47)(33,44)(38,42)(41,46)(43,48);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)(17,20,22,33)(18,24,26,31)(19,28,30,44)(21,32,35,48)(23,36,37,45)(25,38,40,46)(27,41,39,42)(29,43,34,47)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=gh,d^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,f^b=fgh,f^c=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #133110, #133112, #133154, … |
| maximal normal subgroup of | #133110, #133112, #133154, … |
| maximal quotient of | #97020, #97026, #97033, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1515);