Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8479 |
| automorphism | Aut(G) = C23×D4×#120 (order 2048) · Inn(G) = #379 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #379 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | #3559 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
343 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 343 conjugacy classes of subgroups — too many to draw. Only its 71 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,6,8,11)(7,15,16,14)(17,18,21,25,22,26,20,30)(19,27,28,23,29,24,31,32);(1,2)(3,7)(4,11)(5,8)(6,13)(9,14)(10,16)(12,15)(17,19)(18,23)(20,31)(21,28)(22,29)(24,25)(26,32)(27,30);(1,3,5,10)(2,7,8,16)(4,12,13,9)(6,14,11,15)(17,19)(18,24)(20,28)(21,31)(22,29)(23,25)(26,27)(30,32)Polycyclic presentation:
2^8|a^2=f,c^2=gh,d^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=egh,f^b=fgh,d^c=dgh,f^c=fg37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8486, #187227, #187228, … |
| maximal normal subgroup of | C2×#8486, #187227, #187228, … |
| maximal quotient of | #97186, #97189, #97190, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1474);