Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #386 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
381 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 381 conjugacy classes of subgroups — too many to draw. Only its 143 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:
(1,2,4,3)(5,6)(7,14)(8,17)(9,12)(10,13)(11,15)(16,20)(18,19);(5,7,9,15,10,16,8,18)(6,11,12,20,13,19,17,14);(1,3,4,2)(5,6)(7,11)(8,17)(9,12)(10,13)(14,18)(15,20)(16,19)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=f,d^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,d^b=dh,e^b=eh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #94195, #94197, #94199, … |
| maximal normal subgroup of | #94195, #94197, #94199, … |
| maximal quotient of | #109498, #109499, #109500, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3881);