Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8465 |
| automorphism | Aut(G) order 4096 |
| 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 | #3512 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
339 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 339 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) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,4,8,5,6,12,14)(3,9,10,16,11,7,13,15)(17,18)(19,27)(20,31,33,24)(21,34)(22,26)(23,29)(25,37)(28,43,45,48)(30,41)(32,40,36,42)(35,39)(38,44,47,46);(2,6)(4,12)(7,9)(10,13)(17,19)(18,23)(20,28)(21,35)(22,30)(24,38)(25,41)(26,39)(27,34)(29,37)(31,47)(32,46)(33,45)(36,44)(40,43)(42,48);(1,3,5,11)(2,7,6,9)(4,13,12,10)(8,16,14,15)(17,20,22,33)(18,24,26,31)(19,28,30,45)(21,36,37,32)(23,38,39,47)(25,42,34,40)(27,43,41,48)(29,46,35,44)Polycyclic presentation:
2^8|a^2=f,c^2=g,d^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,f^c=fg40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8476, #187106, #187109, … |
| maximal normal subgroup of | C2×#8476, #187106, #187109, … |
| maximal quotient of | #97362, #97365, #97368, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1464);