Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9415 |
| automorphism | Aut(G) = C22×#4933776 (order 2048) · Inn(G) = #386 · Out(G) = C22×D4 |
| 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 | C23 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
190 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 190 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)(3,10,11,14)(4,7,12,13)(5,8)(6,9)(15,16)(17,18)(19,28)(20,32,35,42)(21,36)(22,26)(23,27)(24,30)(25,34,41,33)(29,44,46,38)(31,39)(37,40)(43,45,48,47);(1,3,5,13,6,11,15,7)(2,4,8,14,9,12,16,10)(17,19,21,30,23,31,22,37)(18,24,26,28,27,40,36,39)(20,29,34,47,35,46,33,45)(25,38,42,43,41,44,32,48);(1,4)(2,7)(3,8)(5,14)(6,12)(9,13)(10,15)(11,16)(17,20,23,35)(18,25,27,41)(19,29,31,46)(21,34,22,33)(24,38,40,44)(26,42,36,32)(28,43,39,48)(30,47,37,45)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=g,d^2=gh,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,e^b=egh,f^b=fh,d^c=dgh,e^c=eg,f^c=fgh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9416, #200696, #200697, … |
| maximal normal subgroup of | C2×#9416, #200696, #200697, … |
| maximal quotient of | #100092, #100093, #100096, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2404);