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) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
223 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 223 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,8,6,9,4,14)(3,10,13,15,11,16,12,7)(17,18)(19,26,27,30)(20,23,28,29)(21,24)(22,25)(31,32);(1,3,6,11)(2,7)(4,12,5,13)(8,10)(9,15)(14,16)(17,19,21,29,22,27,31,23)(18,20,24,30,25,28,32,26);(3,11)(7,15)(10,16)(12,13)(17,20)(18,23)(19,24)(21,30)(22,28)(25,29)(26,31)(27,32)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,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 | #132823, #132824, #132827, … |
| maximal normal subgroup of | #132823, #132824, #132827, … |
| maximal quotient of | #100069, #100070, #100071, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2378);