Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | Aut(G) = C23×#72236189 (order 8192) · Inn(G) = #397 · Out(G) = C27 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #397 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
277 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 277 conjugacy classes of subgroups — too many to draw. Only its 103 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) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6,11,13,9,14,10,16)(7,15,12,19,17,8,18,20);(1,2)(3,4)(5,7,9,17)(6,8)(10,12,11,18)(13,19)(14,15)(16,20);(2,4)(5,8,9,15)(6,12)(7,16)(10,19,11,20)(13,17)(14,18)Polycyclic presentation:
2^8|a^2=g,b^2=de,c^2=d,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=dh,e^b=eh,g^b=gh,e^c=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11308, #358347, C4×#2081, … |
| maximal normal subgroup of | C2×#11308, #358347, C4×#2081, … |
| maximal quotient of | #121080, #121096, #121097, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4296);