Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7943 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#95 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
295 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 295 conjugacy classes of subgroups — too many to draw. Only its 115 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:
(2,7,9,20)(3,11)(4,15)(8,18,19,30)(10,24,23,31)(13,27)(14,26)(16,17)(21,22,32,28)(25,29);(1,2,4,10,6,9,17,23)(3,7,13,21,14,20,27,32)(5,8,16,22,12,19,15,28)(11,18,25,31,26,30,29,24);(2,8)(7,18)(9,19)(10,22)(20,30)(21,31)(23,28)(24,32)Polycyclic presentation:
2^8|a^2=e,b^2=f,d^2=gh,f^2=h,b^a=bd,d^a=dg,f^a=fgh,c^b=cg,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #105618, #105619, #106197, … |
| maximal normal subgroup of | #105618, #105619, #106197, … |
| maximal quotient of | #95036, #95038, #95044, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 963);