Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
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 | C24 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
253 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 253 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:
(1,2)(3,4)(5,6,7,8)(10,14)(11,18)(12,19)(15,20)(16,21)(17,24)(22,23);(6,8)(9,10,15,24,12,17,11,14,13,16,18,23,19,22,20,21);(2,4)(10,16)(14,21)(17,22)(23,24)Polycyclic presentation:
2^8|a^2=f,b^2=dfgh,d^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,c^b=ch52 conjugacy classes — view the full character table.
| maximal subgroup of | #130245, #130248, #130249, … |
| maximal normal subgroup of | #130245, #130248, #130249, … |
| maximal quotient of | #137610, #137650, #137975, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6050);