Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24404 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
541 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 541 conjugacy classes of subgroups — too many to draw. Only its 129 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:
(1,2,5,8)(3,7,4,6)(9,10,13,16)(11,14,18,22)(12,15,20,23)(17,24,19,21);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,19)(13,18)(15,21)(16,22)(17,20)(23,24);(1,3,5,4)(2,7,8,6)(9,12)(10,15)(11,17,18,19)(13,20)(14,21,22,24)(16,23);(1,4,5,3)(2,7,8,6)Polycyclic presentation:
2^8|a^2=eh,b^2=e,c^2=ef,d^2=e,b^a=be,c^a=cf,d^a=de,g^a=gh,c^b=cg,f^b=fh,g^c=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | #142665, #142667, #142673, … |
| maximal normal subgroup of | #142665, #142667, #142673, … |
| maximal quotient of | #304177, #304178, #304179, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18936);