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 | #532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
227 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 227 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:
(1,2,6,3)(4,11,12,16)(5,10,15,8)(7,14,9,13)(17,18,22,25)(19,26,28,31)(20,30,21,24)(23,29,32,27);(1,3,6,2)(4,9,12,7)(5,8,15,10)(11,14,16,13)(17,19,22,28)(18,23,25,32)(20,29,21,27)(24,31,30,26);(1,4)(2,7)(3,9)(5,14)(6,12)(8,16)(10,11)(13,15)(19,27)(20,21)(23,31)(24,30)(26,32)(28,29)Polycyclic presentation:
2^8|a^2=h,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×#10126, #236610, #236611, … |
| maximal normal subgroup of | C2×#10126, #236610, #236611, … |
| maximal quotient of | #102508, #102513, #102522, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3114);