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) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | #3147 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
511 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 511 conjugacy classes of subgroups — too many to draw. Only its 191 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:
(2,4)(5,6,9,13,10,14,18,12)(7,15,8,11,16,19,17,20);(5,7,9,8,10,16,18,17)(6,11,12,15,14,20,13,19);(1,2,3,4);(5,8,9,16,10,17,18,7)(6,11,13,19,14,20,12,15)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=f,d^2=g,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #124628, #124632, #128919, … |
| maximal normal subgroup of | #124628, #124632, #128919, … |
| maximal quotient of | #212800, #212801, #212802, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14694);