Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
669 conjugacy classes of subgroups; only the 179 normal subgroups are shown.
This group has 669 conjugacy classes of subgroups — too many to draw. Only its 179 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) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6,7,8)(9,10,13,14)(11,12,16,15)(18,20)(19,23);(10,14)(11,16)(17,18,21,20)(19,22,23,24);(5,6)(7,8)(9,11,13,16)(10,15,14,12)(17,19)(18,22)(20,24)(21,23);(1,2)(3,4)(5,6)(7,8)(9,12,13,15)(10,11,14,16)(17,20,21,18)(19,22,23,24)Polycyclic presentation:
2^8|a^2=efg,b^2=eg,c^2=g,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de52 conjugacy classes — view the full character table.
| maximal subgroup of | #104190, #104206, #104222, … |
| maximal normal subgroup of | #104190, #104206, #104222, … |
| maximal quotient of | #155667, #155739, #155867, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 9622);