Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| 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 | C23×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
595 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 595 conjugacy classes of subgroups — too many to draw. Only its 183 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 — 4 generators on 20 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(14,18)(15,16)(19,20);(5,7,8,11)(6,9,10,12)(13,14,17,19)(15,18,16,20);(6,10)(9,12);(13,15)(16,17)(18,20)Polycyclic presentation:
2^8|a^2=g,b^2=fh,e^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,d^b=de,e^b=eh,e^d=eh70 conjugacy classes — view the full character table.
| maximal subgroup of | #220269, #220271, #220287, … |
| maximal normal subgroup of | #220269, #220271, #220287, … |
| maximal quotient of | #224950, #224974, #224998, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15562);