Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| 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 | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | C2×#95 |
| prime radical | C24 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
273 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 273 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)(3,9)(4,12)(5,8)(6,11)(7,13)(10,15)(14,16)(17,18,20,24,22,25,21,30)(19,26,28,32,27,23,29,31);(1,3)(2,6,8,14)(4,11)(5,10)(7,9,12,15)(13,16)(17,19,21,29,22,27,20,28)(18,23,24,31,25,26,30,32);(3,10)(6,14)(9,15)(11,16)(19,27)(23,26)(28,29)(31,32)Polycyclic presentation:
2^8|a^2=e,b^2=f,d^2=g,e^2=h,f^2=h,b^a=bd,d^a=dg,f^a=fgh,c^b=cgh,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #105796, #105797, #105798, … |
| maximal normal subgroup of | #105796, #105797, #105798, … |
| maximal quotient of | #95912, #95913, #95914, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1077);