Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
535 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10,14,17,15,19,12,22)(11,20,18,13,23,24,21,16);(5,6)(7,8)(9,11,15,23)(10,16,19,13)(12,18,14,21)(17,24,22,20);(2,4)(6,8)(9,12,15,14)(10,17,19,22)(11,21,23,18)(13,24,16,20);(2,4)(9,13,15,16)(10,18,19,21)(11,22,23,17)(12,24,14,20)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=h,d^2=h,e^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,e^b=eh,e^d=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #6632144, #6632145, #6632176, … |
| maximal normal subgroup of | #6632144, #6632145, #6632176, … |
| maximal quotient of | #333421, #333436, #333450, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23455);