Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C23 |
| commutator, socle | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
499 conjugacy classes of subgroups; only the 203 normal subgroups are shown.
This group has 499 conjugacy classes of subgroups — too many to draw. Only its 203 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:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,15,17,20,16,19);(6,9)(10,12)(13,15,18,20,17,19,16,14);(1,2,3,4)(13,16,17,18)(14,15,20,19);(5,7,8,11)(6,10,9,12)(13,16,17,18)(14,19,20,15)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=f,d^2=e,g^2=h,b^a=be,c^a=cf,c^b=ch88 conjugacy classes — view the full character table.
| maximal subgroup of | #124849, #124972, #183845, … |
| maximal normal subgroup of | #124849, #124972, #183845, … |
| maximal quotient of | #206514, #206515, #206516, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14253);