Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C2×#135 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
326 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 326 conjugacy classes of subgroups — too many to draw. Only its 159 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,7,5,3,6,8)(9,10,14,17)(11,18,20,23)(12,16,13,21)(15,22,24,19);(9,11,14,20)(10,15,17,24)(12,19,13,22)(16,23,21,18);(1,3,4,8,5,2,6,7);(2,3)(7,8)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=fg,d^2=g,e^2=g,f^2=h,b^a=be,d^a=dh,e^a=eg,e^b=eg,d^c=dh70 conjugacy classes — view the full character table.
| maximal subgroup of | #136485, #137026, #137028, … |
| maximal normal subgroup of | #136485, #137026, #137028, … |
| maximal quotient of | #183179, #183672, #183949, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11027);