Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8491 |
| automorphism | Aut(G) = C22×#648360 (order 2048) · Inn(G) = #379 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #379 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | #3554 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
336 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 69 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:
(1,2,5,8,6,10,4,9)(3,11,12,7,13,15,14,16)(17,18)(19,27,30,23)(20,31,33,40)(21,26)(22,25)(24,32,38,34)(28,43)(29,42,47,37)(35,39)(36,44)(41,46)(45,48);(1,3)(2,7)(4,14)(5,12)(6,13)(8,15)(9,11)(10,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,36)(25,27)(26,37)(31,48)(32,44)(33,45)(34,46)(35,47)(38,41)(39,42)(40,43);(2,8)(4,5)(7,15)(9,10)(11,16)(12,14)(17,20,22,33)(18,24,25,38)(19,28,30,45)(21,34,35,32)(23,36,27,41)(26,40,39,31)(29,46,47,44)(37,43,42,48)Polycyclic presentation:
2^8|a^2=f,c^2=gh,d^2=h,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,e^c=eg,f^c=fg34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8503, #187508, #187510, … |
| maximal normal subgroup of | C2×#8503, #187508, #187510, … |
| maximal quotient of | #96981, #96983, #96987, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1491);