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 | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
550 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)(9,10,14,15,13,16,12,18)(11,19,22,17,21,24,20,23);(5,6)(7,8)(10,15)(11,20)(12,14)(16,18)(17,23)(21,22);(2,4)(6,8)(10,16)(15,18)(17,23)(19,24);(2,4)(9,11,13,21)(10,17,16,23)(12,22,14,20)(15,19,18,24)Polycyclic presentation:
2^8|a^2=eh,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 | #6632147, #6632150, #6632179, … |
| maximal normal subgroup of | #6632147, #6632150, #6632179, … |
| maximal quotient of | #333416, #333431, #333445, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23457);