Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C22×D4 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
501 conjugacy classes of subgroups; only the 247 normal subgroups are shown.
This group has 501 conjugacy classes of subgroups — too many to draw. Only its 247 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)(15,17)(16,19)(18,20);(1,3,4,2)(14,17)(15,20)(18,19);(6,9)(10,12);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^8|a^2=g,b^2=g,d^2=f,e^2=h,b^a=be,c^a=cf,e^a=eh,e^b=eh76 conjugacy classes — view the full character table.
| maximal subgroup of | #109827, #109829, #109955, … |
| maximal normal subgroup of | #109827, #109829, #109955, … |
| maximal quotient of | #207292, #207304, #207316, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14346);