Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10238 |
| automorphism | Aut(G) order 8192 |
| 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 | C23 | #120 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
252 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 61 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)(3,11)(4,8)(5,14)(6,10)(7,12)(9,15)(13,16)(17,18,22,26)(19,27,29,32)(20,25,21,24)(23,30,31,28);(1,3,6,4)(2,7,10,16)(5,13,15,12)(8,14,11,9)(17,19,22,29)(18,23,26,31)(20,30,21,28)(24,32,25,27);(1,4,6,3)(2,8,10,11)(5,12,15,13)(7,9,16,14)(18,24)(20,21)(23,27)(25,26)(28,30)(31,32)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=gh,d^2=g,e^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,c^b=cf,d^b=dg,d^c=dg,e^c=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10253, #237566, #237570, … |
| maximal normal subgroup of | C2×#10253, #237566, #237570, … |
| maximal quotient of | #102534, #102537, #102539, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3241);