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 | C22×D4 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
268 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 268 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)(3,7)(4,6)(5,9)(8,12)(10,15)(11,16)(13,14)(17,18,22,26)(19,27,29,32)(20,25,21,24)(23,30,31,28);(2,6)(4,12)(7,15)(8,9)(10,13)(14,16)(17,19)(18,23)(20,30)(21,28)(22,29)(24,32)(25,27)(26,31);(1,3,5,11)(2,7,9,16)(4,13,12,10)(6,14,8,15)(18,24)(20,21)(23,27)(25,26)(28,30)(31,32)Polycyclic presentation:
2^8|a^2=h,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×#10249, #237566, #237569, … |
| maximal normal subgroup of | C2×#10249, #237566, #237569, … |
| maximal quotient of | #102533, #102535, #102536, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3237);