Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20671 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#117 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C2 | C2×#432 |
| prime radical | C24 | M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ #405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
262 conjugacy classes of subgroups; only the 157 normal subgroups are shown.
This group has 262 conjugacy classes of subgroups — too many to draw. Only its 157 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 — 4 generators on 32 points:
(1,2,4,9,5,10,17,25,6,7,15,20,18,21,30,32)(3,11,13,27,14,19,29,24,12,26,16,31,28,8,22,23);(2,7)(4,15)(8,19)(10,21)(11,26)(13,16)(17,30)(22,29);(1,3)(2,8,7,19)(4,16)(5,14)(6,12)(9,24,20,23)(10,11,21,26)(13,15)(17,22)(18,28)(25,31,32,27)(29,30);(3,12)(8,19)(11,26)(13,16)(14,28)(22,29)(23,24)(27,31)Polycyclic presentation:
2^8|a^2=f,c^2=e,f^2=g,g^2=h,b^a=be,c^a=cg,e^a=eh,f^b=fh,d^c=dh,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #137786, #137787, #137816, … |
| maximal normal subgroup of | #137786, #137787, #137816, … |
| maximal quotient of | #492927, #492960, #493110, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26415);