Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10524 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #394 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | #3253 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
185 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 185 conjugacy classes of subgroups — too many to draw. Only its 45 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)(3,7)(4,6)(5,9)(8,12)(10,13)(11,14)(15,16)(18,23)(19,26)(20,21)(24,27)(25,30)(28,29);(2,6)(3,10)(4,12)(7,14)(8,9)(11,16)(17,18)(19,27)(20,30)(21,23)(22,25)(24,29)(26,32)(28,31);(1,3,4,10,5,11,12,16)(2,7,8,15,9,14,6,13)(17,19,21,26,22,29,20,28)(18,24,30,32,25,27,23,31)Polycyclic presentation:
2^8|c^2=dh,d^2=g,e^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cf,d^b=dg,f^b=fgh,e^c=egh,f^c=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | #94678, #151252, C2×#10525, … |
| maximal normal subgroup of | #94678, #151252, C2×#10525, … |
| maximal quotient of | #104878, #104885, #104894, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3513);