Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
531 conjugacy classes of subgroups; only the 211 normal subgroups are shown.
This group has 531 conjugacy classes of subgroups — too many to draw. Only its 211 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) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6,8,7)(9,10,13,16)(11,17,18,24)(12,20,21,15)(14,22,23,19);(9,11,12,19,13,18,21,22)(10,14,15,17,16,23,20,24);(1,2,3,4);(5,7,8,6)(11,18)(14,23)(17,24)(19,22)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=f,d^2=gh,e^2=h,b^a=be,c^a=cf,e^a=eh,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #100096, #100106, #124630, … |
| maximal normal subgroup of | #100096, #100106, #124630, … |
| maximal quotient of | #212801, #212809, #212818, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14679);