Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3186 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | #3147 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
423 conjugacy classes of subgroups; only the 155 normal subgroups are shown.
This group has 423 conjugacy classes of subgroups — too many to draw. Only its 155 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)(4,7)(5,8)(9,10,14,18,15,19,13,23)(11,16,20,24,21,17,12,22);(1,3,6,2)(4,7,5,8)(9,11,14,20,15,21,13,12)(10,16,18,24,19,17,23,22);(1,4,6,5)(2,7,3,8)(9,12,13,21,15,20,14,11)(10,17,18,22,19,16,23,24);(1,5,6,4)(2,7,3,8)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=ef,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce64 conjugacy classes — view the full character table.
| maximal subgroup of | #137904, #139104, #139105, … |
| maximal normal subgroup of | #137904, #139104, #139105, … |
| maximal quotient of | #219360, #219361, #219362, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15268);