Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
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 | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
495 conjugacy classes of subgroups; only the 247 normal subgroups are shown.
This group has 495 conjugacy classes of subgroups — too many to draw. Only its 247 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:
(2,4)(5,6,8,7)(9,10,13,17)(11,15,18,22)(12,14,20,16)(19,21,24,23);(5,7,8,6)(10,14)(12,20)(15,21)(16,17)(19,24)(22,23);(1,2,3,4)(9,11)(10,15)(12,19)(13,18)(14,21)(16,23)(17,22)(20,24);(11,18)(15,22)(19,24)(21,23)Polycyclic presentation:
2^8|a^2=g,b^2=gh,c^2=f,e^2=h,b^a=be,c^a=cf,e^a=eh,e^b=eh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #367504, #367506, #367540, … |
| maximal normal subgroup of | #367504, #367506, #367540, … |
| maximal quotient of | #203620, #203630, #203640, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14162);