Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18376 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
598 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 598 conjugacy classes of subgroups — too many to draw. Only its 115 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) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(6,8)(9,10)(11,17)(12,20)(13,15)(14,19)(16,22)(18,24)(21,23);(1,2)(3,8)(4,6)(5,7)(9,11)(10,14)(12,21)(13,18)(15,23)(16,24)(17,20)(19,22);(3,7)(5,8);(2,6)(5,8)(10,15)(11,18)(16,20)(19,21)Polycyclic presentation:
2^8|e^2=g,b^a=be,c^a=cf,d^a=dgh,e^a=eg,d^b=dg,e^b=eg,f^b=fh,e^c=eh43 conjugacy classes — view the full character table.
| maximal subgroup of | #486097, #486101, #486109, … |
| maximal normal subgroup of | #486097, #486101, #486109, … |
| maximal quotient of | #169613, #169625, #169677, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11364);