Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9495 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C24 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
279 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 279 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:
(1,2,6,11,5,3,8,14)(4,9,16,25,15,12,20,27)(7,10,21,22,18,13,23,19)(17,26,30,31,28,24,32,29);(1,3,6,19,5,2,8,22)(4,12,20,31,15,9,16,29)(7,13,21,14,18,10,23,11)(17,24,32,25,28,26,30,27);(1,4,7,17)(2,9,10,26)(3,12,13,24)(5,15,18,28)(6,20,21,32)(8,16,23,30)(11,27,22,29)(14,25,19,31)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dg,e^b=eh,f^b=fg,d^c=dh,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #132821, #132823, #132825, … |
| maximal normal subgroup of | #132821, #132823, #132825, … |
| maximal quotient of | #100188, #100190, #100200, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2500);