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 | #532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
210 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 210 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) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,10,11,14)(4,7,12,13)(5,8)(6,9)(15,16)(17,18)(19,22,26,31)(20,23)(21,24)(25,28)(27,29,32,30);(1,3,5,13,6,11,15,7)(2,4,8,14,9,12,16,10)(17,19)(18,22,24,31)(20,27)(21,26)(23,29,25,30)(28,32);(1,4)(2,7)(3,8)(5,14)(6,12)(9,13)(10,15)(11,16)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31)Polycyclic presentation:
2^8|a^2=f,b^2=df,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×#9415, #200696, #200697, … |
| maximal normal subgroup of | C2×#9415, #200696, #200697, … |
| maximal quotient of | #100091, #100094, #100095, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2403);