Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10120 |
| automorphism | Aut(G) = C2×#72278177 (order 4096) · Inn(G) = #392 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | #3214 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
238 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 49 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(3,9)(4,11)(6,14)(7,12)(10,13)(15,16)(17,18)(19,26)(20,30)(21,24)(22,25)(23,27)(28,31)(29,32);(1,2,5,8)(3,6,10,15)(4,12,11,7)(9,16,13,14)(17,19,22,28)(18,23,25,32)(20,29,21,27)(24,31,30,26);(1,3)(2,6)(4,13)(5,10)(7,16)(8,15)(9,11)(12,14)(19,27)(20,21)(23,31)(24,30)(26,32)(28,29)Polycyclic presentation:
2^8|b^2=h,d^2=g,e^2=gh,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=egh,c^b=cf,d^b=dg,e^b=eh,f^b=fg,d^c=dh,e^c=egh,f^c=fg31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10122, #236607, #236611, … |
| maximal normal subgroup of | C2×#10122, #236607, #236611, … |
| maximal quotient of | #102507, #102509, #102519, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3110);