Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13247 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1972 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#509 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
207 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 207 conjugacy classes of subgroups — too many to draw. Only its 65 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,7,12)(3,5,11,23)(4,16)(6,25,24,10)(8,15,22,13)(9,26)(14,21)(17,20)(18,27)(19,28)(29,30)(31,32);(1,3,6,13,8,10,5,12,7,11,24,15,22,25,23,2)(4,17,26,9,20,16,30,27,21,31,19,28,32,14,18,29);(1,4,6,19,8,20,5,18,7,21,24,26,22,32,23,30)(2,9,11,16,13,27,25,31,12,28,3,14,15,29,10,17)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=e,d^2=gh,e^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,c^b=cg,e^b=eh,f^c=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13247, #495237, #495899, … |
| maximal normal subgroup of | C2×#13247, #495237, #495899, … |
| maximal quotient of | #140746, #140747, #140781, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6235);