Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9345 |
| automorphism | Aut(G) = C23×D4×#120 (order 2048) · Inn(G) = #386 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
208 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 208 conjugacy classes of subgroups — too many to draw. Only its 61 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:
(1,2,4,7,6,10,5,9)(3,11,12,8,13,15,14,16)(17,18,20,24,21,23,25,31)(19,22,27,29,26,30,28,32);(2,7,10,9)(8,15,16,11)(17,19,25,27,21,26,20,28)(18,22,24,32,23,30,31,29);(1,3)(2,8)(4,12)(5,14)(6,13)(7,15)(9,11)(10,16)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=gh,e^2=gh,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=egh,d^b=dh,f^b=fh,e^c=egh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9351, #199864, #199865, … |
| maximal normal subgroup of | C2×#9351, #199864, #199865, … |
| maximal quotient of | #100015, #100017, #100019, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2339);