Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | Aut(G) order 16384 |
| 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 | #3150 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
507 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 507 conjugacy classes of subgroups — too many to draw. Only its 187 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)(5,6,10,14,11,15,9,19)(7,16,8,13,17,20,18,12);(5,7,9,18,11,17,10,8)(6,12,14,16,15,13,19,20);(1,2,3,4)(7,17)(8,18)(12,13)(16,20);(5,8,10,17,11,18,9,7)(6,13,14,20,15,12,19,16)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,c^b=ch,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #124786, #124790, #124794, … |
| maximal normal subgroup of | #124786, #124790, #124794, … |
| maximal quotient of | #212967, #212968, #212969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14698);