Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
400 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 83 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)(5,6,8,10,9,12,7,11)(13,14)(15,19,20,17)(16,18);(1,2,3,4)(6,10)(7,8)(11,12)(13,14)(15,17)(16,18)(19,20);(13,15,16,20)(14,17,18,19)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=h,d^2=g,f^2=g,b^a=bd,c^a=ce,e^a=eh,d^b=dg,e^b=eh,f^b=fg,d^c=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #131458, #131460, #131461, … |
| maximal normal subgroup of | #131458, #131460, #131461, … |
| maximal quotient of | #99549, #99564, #99633, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2003);