Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9369 |
| automorphism | Aut(G) = C23×#2403 (order 1024) · Inn(G) = #386 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | #121 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
195 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 195 conjugacy classes of subgroups — too many to draw. Only its 57 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)(3,6,10,16)(4,9)(5,8)(7,11,15,12)(13,14)(17,18)(19,27)(20,31,33,24)(21,34)(22,26)(23,29)(25,36)(28,41,44,48)(30,42)(32,39,35,40)(37,43,47,45)(38,46);(1,3)(2,6,8,16)(4,12)(5,10)(7,14,15,9)(11,13)(17,19,21,29,22,30,36,46)(18,23,25,27,26,38,34,42)(20,28,35,45,33,44,32,43)(24,37,40,41,31,47,39,48);(1,3)(2,7)(4,12)(5,10)(6,14)(8,15)(9,16)(11,13)(17,20)(18,24)(19,28)(21,35)(22,33)(23,37)(25,40)(26,31)(27,41)(29,45)(30,44)(32,36)(34,39)(38,47)(42,48)(43,46)Polycyclic presentation:
2^8|a^2=f,b^2=df,d^2=gh,e^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,d^b=dh,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fg37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9369, #200118, #200119, … |
| maximal normal subgroup of | C2×#9369, #200118, #200119, … |
| maximal quotient of | #100043, #100044, #100047, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2357);