Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
443 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 443 conjugacy classes of subgroups — too many to draw. Only its 175 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 — 4 generators on 20 points:
(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,14,19,15)(16,18,20,17);(6,9)(10,12)(13,15,18,20,19,14,17,16);(5,7,8,11)(6,10,9,12)(13,16,18,15,19,20,17,14);(1,3,4,2)(13,17,19,18)(14,16,15,20)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=e,d^2=g,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=ch,d^b=dh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #100111, #100117, #209163, … |
| maximal normal subgroup of | #100111, #100117, #209163, … |
| maximal quotient of | #212615, #212621, #212627, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14643);