Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | Aut(G) = C23×D4×#120 (order 2048) · Inn(G) = C2×C22⋊C4 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
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 | C4×C22⋊C4 |
| prime radical | #121 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
200 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 81 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)(17,18,22,25,23,19,21,27)(20,24,29,31,30,26,28,32);(1,3)(2,6,8,14)(4,11)(5,10)(7,9,12,15)(13,16)(17,19,23,18)(20,26,30,24)(21,25,22,27)(28,31,29,32);(3,10)(6,14)(9,15)(11,16)(17,20,22,29,23,30,21,28)(18,24,25,31,19,26,27,32)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=e,d^2=g,e^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 | C2×#8250, #182925, #182928, … |
| maximal normal subgroup of | C2×#8250, #182925, #182928, … |
| maximal quotient of | #96780, #96781, #96784, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1238);