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) = C25 |
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 | |
923 conjugacy classes of subgroups; only the 227 normal subgroups are shown.
This group has 923 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:
(2,4)(6,7)(9,10,13,11)(12,15,16,14)(18,21)(22,24);(5,6,8,7)(10,11)(14,15)(17,18)(19,22)(20,21)(23,24);(1,2)(3,4)(5,7,8,6)(9,11,13,10)(12,15,16,14)(17,19,20,23)(18,22,21,24);(9,12)(10,14)(11,15)(13,16)(17,18)(19,22)(20,21)(23,24)Polycyclic presentation:
2^8|a^2=g,b^2=eh,c^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg64 conjugacy classes — view the full character table.
| maximal subgroup of | #101077, #101080, #101109, … |
| maximal normal subgroup of | #101077, #101080, #101109, … |
| maximal quotient of | #154132, #154205, #154207, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8002);