Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32898 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
657 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 657 conjugacy classes of subgroups — too many to draw. Only its 135 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,11)(12,15,20,17)(13,16,21,18)(19,23,24,22);(2,5)(6,8)(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22);(1,3,4,7)(2,6,5,8)(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,23)(21,24);(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=e,b^a=be,c^a=cf,d^a=dg,g^c=gh,f^d=fh49 conjugacy classes — view the full character table.
| maximal subgroup of | #246329, #246330, #246333, … |
| maximal normal subgroup of | #246329, #246330, #246333, … |
| maximal quotient of | #349832, #349833, #349834, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26011);