Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10012 |
| automorphism | Aut(G) = C22×#72210469 (order 4096) · Inn(G) = #127 · Out(G) = C22×#120 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #399 |
| prime radical | C23 | #123 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
148 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 148 conjugacy classes of subgroups — too many to draw. Only its 53 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:
(1,2,5,7,6,10,4,9)(3,11,13,16,14,15,12,8)(17,18,23,27,22,28,21,26)(19,29,37,24,32,40,31,39)(20,25,35,44,36,43,34,42)(30,45,33,48,47,41,46,38);(2,7)(3,12)(4,5)(8,16)(9,10)(13,14)(17,19,21,37,22,32,23,31)(18,24,26,40,28,39,27,29)(20,33,34,47,36,46,35,30)(25,41,42,38,43,45,44,48);(1,3,6,14)(2,8,10,16)(4,13,5,12)(7,15,9,11)(17,20,22,36)(18,25,28,43)(19,30,32,47)(21,34,23,35)(24,38,39,48)(26,42,27,44)(29,41,40,45)(31,46,37,33)Polycyclic presentation:
2^8|a^2=dg,b^2=de,c^2=g,d^2=g,e^2=h,f^2=g,b^a=bd,c^a=ce,f^a=fgh,c^b=cf,d^b=dg,e^b=eg,f^b=fg,d^c=dh,e^c=eh,f^c=fg40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10669, #243890, #243892, … |
| maximal normal subgroup of | C2×#10669, #243890, #243892, … |
| maximal quotient of | #105233, #105234, #105237, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3657);