Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9369 |
| automorphism | Aut(G) = C23×#35346 (order 2048) · Inn(G) = #386 · Out(G) = C25 |
| 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 | C22×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
246 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 246 conjugacy classes of subgroups — too many to draw. Only its 59 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,12,18,28,3,15)(4,20,23,30,32,11,19,27)(6,13,16,8,7,14,17,10)(9,26,24,22,29,31,25,21);(2,8,13,12)(3,16)(4,21)(5,17)(9,24,20,30)(10,14,15,28)(11,27,29,25)(19,23)(22,32)(26,31);(2,9)(3,17)(4,22)(5,16)(8,24)(10,25)(11,14)(12,30)(13,20)(15,27)(19,23)(21,32)(26,31)(28,29)Polycyclic presentation:
2^8|a^2=f,b^2=df,d^2=gh,e^2=h,f^2=gh,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×#9379, #200318, #200322, … |
| maximal normal subgroup of | C2×#9379, #200318, #200322, … |
| maximal quotient of | #100043, #100045, #100051, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2367);