Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33752 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
318 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 318 conjugacy classes of subgroups — too many to draw. Only its 103 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)(10,14)(11,17)(13,21)(15,23)(16,20)(18,22)(19,24);(1,3,4,8)(2,5,6,7)(9,10,12,14)(11,18,19,15)(13,16,20,21)(17,23,24,22);(3,8)(5,7)(9,11)(10,15)(12,19)(13,22)(14,18)(16,24)(17,21)(20,23);(2,6)(5,7)(9,10,12,14)(11,15,19,18)(13,16,20,21)(17,22,24,23)Polycyclic presentation:
2^8|b^2=e,d^2=h,f^2=gh,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fg,g^a=gh,c^b=ce,f^c=fg,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #139086, #139092, #139105, … |
| maximal normal subgroup of | #139086, #139092, #139105, … |
| maximal quotient of | #496352, #496480, #496736, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26761);