Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | C2×#95 |
| prime radical | C24 | C42 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
275 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 275 conjugacy classes of subgroups — too many to draw. Only its 123 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)(3,11,13,7)(4,8)(5,10)(6,9)(12,16,14,15)(17,18,21,24,22,25,20,29)(19,26,27,31,28,23,30,32);(1,3,5,14,6,13,4,12)(2,7,10,16,9,11,8,15)(17,19)(18,23)(20,27)(21,30)(22,28)(24,31)(25,26)(29,32);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,26,32)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=g,e^2=h,f^2=gh,b^a=bd,c^a=ch,d^a=dg,e^b=eg64 conjugacy classes — view the full character table.
| maximal subgroup of | #106000, #106001, #106068, … |
| maximal normal subgroup of | #106000, #106001, #106068, … |
| maximal quotient of | #94919, #94920, #94931, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 924);