Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11502 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C24 | C22×C4 |
| Frattini, cosocle | C22×D4 | C23 |
| prime radical | C2×#120 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
300 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 300 conjugacy classes of subgroups — too many to draw. Only its 65 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) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6)(7,10)(8,15,16,11)(9,12)(13,19)(14,17,20,18);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,19)(15,17)(16,20);(6,11)(7,13)(10,17)(12,15)(14,20)(18,19)Polycyclic presentation:
2^8|a^2=f,b^2=d,b^a=bd,c^a=ce,e^a=eg,g^a=gh,c^b=ch,f^c=fg,f^e=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #130268, #130270, #130272, … |
| maximal normal subgroup of | #130268, #130270, #130272, … |
| maximal quotient of | #134228, #134259, #134282, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5783);