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) = C23×C42 |
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 | |
669 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 669 conjugacy classes of subgroups — too many to draw. Only its 175 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)(11,15)(13,16)(18,21)(22,23);(5,6,7,8)(17,18,20,21)(19,22,24,23);(1,2)(3,4)(5,6)(7,8)(9,10,12,14)(11,13,15,16)(17,18)(19,23)(20,21)(22,24);(9,11)(10,13)(12,15)(14,16)(17,19,20,24)(18,22,21,23)Polycyclic presentation:
2^8|b^2=g,c^2=h,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 | #255259, #255279, #255309, … |
| maximal normal subgroup of | #255259, #255279, #255309, … |
| maximal quotient of | #155814, #155831, #155865, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10048);