Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C25 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
891 conjugacy classes of subgroups; only the 227 normal subgroups are shown.
This group has 891 conjugacy classes of subgroups — too many to draw. Only its 227 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:
(6,8)(9,10,12,13)(11,15,16,14)(19,23)(21,24);(2,4)(6,8)(10,13)(14,15)(17,18,20,22)(19,21,23,24);(1,2)(3,4)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24);(9,11,12,16)(10,14,13,15)(17,19,20,23)(18,21,22,24)Polycyclic presentation:
2^8|a^2=e,b^2=eh,d^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg64 conjugacy classes — view the full character table.
| maximal subgroup of | #394455, #394456, #394457, … |
| maximal normal subgroup of | #394455, #394456, #394457, … |
| maximal quotient of | #154186, #154383, #154389, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8204);