Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8101 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#95 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C24 | C42 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
260 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 260 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,10,12,7)(4,14)(5,8)(6,9)(11,15,13,16)(17,18,21,25,22,26,20,24)(19,27,30,31,29,23,28,32);(1,3,5,13,6,12,4,11)(2,7,8,15,9,10,14,16)(17,19)(18,23,26,27)(20,28)(21,30)(22,29)(24,31,25,32);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,31,27,32)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=gh,e^2=h,f^2=g,b^a=bd,c^a=cg,d^a=dg,f^a=fh,d^b=dh,e^b=eg52 conjugacy classes — view the full character table.
| maximal subgroup of | #105882, #105944, #106897, … |
| maximal normal subgroup of | #105882, #105944, #106897, … |
| maximal quotient of | #95153, #95157, #95160, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1110);