Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9471 |
| automorphism | Aut(G) order 8192 |
| 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 | C25 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
307 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 307 conjugacy classes of subgroups — too many to draw. Only its 61 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:
(1,2,5,10,6,11,4,17)(3,12,16,23,13,27,14,8)(7,9,19,25,20,21,18,26)(15,28,30,22,29,32,31,24);(1,3,7,15)(2,8,21,22)(4,14,18,31)(5,16,19,30)(6,13,20,29)(9,24,11,23)(10,12,26,32)(17,27,25,28);(2,9)(3,13)(4,18)(5,19)(8,22)(11,21)(12,27)(14,30)(15,29)(16,31)(23,24)(28,32)Polycyclic presentation:
2^8|a^2=f,b^2=dfh,d^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cg,e^b=eh,d^c=dh,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #132895, #132896, #132899, … |
| maximal normal subgroup of | #132895, #132896, #132899, … |
| maximal quotient of | #100167, #100169, #100175, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2469);