Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10078 |
| automorphism | Aut(G) = C27×#120 (order 4096) · Inn(G) = #120 · Out(G) = C27 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
230 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,3)(5,6)(7,11)(8,10)(9,12)(15,20)(17,19);(1,2)(3,4)(5,6,7,11)(8,12,9,10)(13,14)(15,17)(16,18)(19,20);(1,3,4,2)(6,10)(8,9)(11,12)(14,17)(15,20)(18,19)Polycyclic presentation:
2^8|b^2=g,c^2=dg,e^2=g,f^2=gh,b^a=bd,c^a=ce,e^a=eg,f^a=fh,c^b=cf,e^b=eh,f^b=fgh,e^c=eg,f^c=fgh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10427, #238557, #238560, … |
| maximal normal subgroup of | C2×#10427, #238557, #238560, … |
| maximal quotient of | #104829, #104830, #104832, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3415);