Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9385 |
| 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 | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | #392 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
248 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 248 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:
(1,2,6,11,7,9,26,27)(3,13,10,28,15,4,18,14)(5,21,23,30,8,12,25,16)(17,20,24,31,22,19,29,32);(1,3,14,21,8,17,32,9)(2,7,15,28,12,5,22,31)(4,11,23,18,19,16,26,29)(6,24,13,27,25,10,20,30);(2,9)(3,15)(4,19)(6,25)(10,29)(11,30)(12,21)(13,20)(16,27)(17,22)(18,24)(23,26)Polycyclic presentation:
2^8|a^2=f,b^2=df,d^2=gh,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cg,d^b=dh,e^b=eh,f^b=fh,d^c=dh,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #132821, #132822, #132825, … |
| maximal normal subgroup of | #132821, #132822, #132825, … |
| maximal quotient of | #100067, #100068, #100073, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2377);