Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
423 conjugacy classes of subgroups; only the 171 normal subgroups are shown.
This group has 423 conjugacy classes of subgroups — too many to draw. Only its 171 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:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,22,20,23,19,24);(1,3,4,8)(2,5,6,7)(17,19,20,21)(18,22,23,24);(3,8)(5,7)(10,13)(14,16);(2,6)(5,7)(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^8|a^2=g,b^2=e,d^2=f,g^2=h,b^a=be,c^a=cf,d^a=deh,c^b=ceh64 conjugacy classes — view the full character table.
| maximal subgroup of | #161264, #161265, #161272, … |
| maximal normal subgroup of | #161264, #161265, #161272, … |
| maximal quotient of | #355293, #355379, #355500, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15002);