Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9519 |
| 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 | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
269 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 269 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:
(1,2,5,12,6,13,15,25)(3,8,7,14,17,10,22,29)(4,18,21,24,20,30,31,28)(9,26,27,32,11,16,23,19);(2,8,13,10)(4,19)(9,23,11,27)(12,14,25,29)(16,31)(18,28,30,24)(20,32)(21,26);(2,9)(3,15)(4,20)(5,17)(8,23)(10,27)(11,13)(12,24)(14,18)(16,21)(19,32)(25,28)(26,31)(29,30)Polycyclic presentation:
2^8|a^2=f,b^2=dfh,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=egh,e^b=eh,d^c=dh,e^c=eg,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9525, #201993, #201994, … |
| maximal normal subgroup of | C2×#9525, #201993, #201994, … |
| maximal quotient of | #100223, #100225, #100231, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2513);