Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D8 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#466 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
291 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 291 conjugacy classes of subgroups — too many to draw. Only its 83 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 — 3 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,13,16,12,20,21,24,14,17,23,22,15,11,18,19);(1,2,3,4)(9,11)(10,15)(12,17)(13,22)(14,20)(16,23)(18,19)(21,24);(6,8)(9,12,14,15)(10,11,17,20)(13,21,23,18)(16,19,22,24)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=h,d^2=gh,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^b=dg,f^b=fg,g^b=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #131332, #131334, #131410, … |
| maximal normal subgroup of | #131332, #131334, #131410, … |
| maximal quotient of | #136604, #136632, #136879, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5980);