Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C25 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
947 conjugacy classes of subgroups; only the 255 normal subgroups are shown.
This group has 947 conjugacy classes of subgroups — too many to draw. Only its 255 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,19)(20,23)(21,24);(6,8)(9,11,13,19,14,20,12,21)(10,15,16,18,17,24,22,23);(2,4);(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,24,18)Polycyclic presentation:
2^8|a^2=g,b^2=eg,d^2=h,e^2=h,b^a=be,c^a=cf,e^a=eh,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #108027, #108123, #110476, … |
| maximal normal subgroup of | #108027, #108123, #110476, … |
| maximal quotient of | #195811, #195823, #195835, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13960);