Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
403 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 403 conjugacy classes of subgroups — too many to draw. Only its 143 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)(9,10,12,14)(11,15,16,13)(17,18,20,21)(19,22,24,23);(1,3,5,4)(2,6,8,7)(17,19)(18,22)(20,24)(21,23);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,13,14,15)(18,21)(19,23)(22,24);(1,4,5,3)(2,7,8,6)Polycyclic presentation:
2^8|a^2=f,b^2=e,c^2=efh,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #305272, #305280, #305322, … |
| maximal normal subgroup of | #305272, #305280, #305322, … |
| maximal quotient of | #360805, #362102, #365812, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19098);