Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| 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 | C2×#449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
555 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)(9,10,14,17,15,13,12,21)(11,18,19,22,20,24,23,16);(5,6)(7,8)(9,11,15,20)(10,16,13,22)(12,19,14,23)(17,24,21,18);(2,4)(6,8)(9,12,15,14)(10,17,13,21)(11,19,20,23)(16,24,22,18);(2,4)(9,13,15,10)(11,18,20,24)(12,21,14,17)(16,23,22,19)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=def,c^b=cg,d^b=de,e^b=eh,e^d=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #6632133, #6632134, #6632165, … |
| maximal normal subgroup of | #6632133, #6632134, #6632165, … |
| maximal quotient of | #333285, #333300, #333314, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23439);