Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
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 | #3169 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
357 conjugacy classes of subgroups; only the 155 normal subgroups are shown.
This group has 357 conjugacy classes of subgroups — too many to draw. Only its 155 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:
(1,2,4,6)(3,5,8,7)(9,10,14,17,13,16,22,24)(11,18,20,21,19,15,23,12);(2,5)(3,8)(6,7)(9,11)(10,15)(12,17)(13,19)(14,20)(16,18)(21,24)(22,23);(2,6)(5,7)(9,12,14,18,13,21,22,15)(10,11,17,20,16,19,24,23);(11,19)(12,21)(15,18)(20,23)Polycyclic presentation:
2^8|a^2=f,c^2=fg,e^2=g,f^2=h,b^a=be,c^a=cg,e^a=eg,d^b=dh,e^b=eg,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #136567, #137501, #137503, … |
| maximal normal subgroup of | #136567, #137501, #137503, … |
| maximal quotient of | #182567, #183694, #183942, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11061);