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) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C24 | C22×C4 |
| Frattini, prime radical, cosocle | C22×D4 | C23 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
256 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 256 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) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6)(7,13)(8,16,17,11)(9,12)(10,15,18,20)(14,19);(1,2,3,4)(7,14)(10,18)(13,19)(15,20);(5,7,9,14)(6,10,12,18)(8,15,17,20)(11,13,16,19)Polycyclic presentation:
2^8|a^2=f,b^2=d,c^2=h,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 | #130269, #130271, #130273, … |
| maximal normal subgroup of | #130269, #130271, #130273, … |
| maximal quotient of | #134229, #134259, #134283, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5784);