Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24398 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
348 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,8)(4,6)(5,7)(9,10,15,17,14,12,13,18)(11,19,23,20,22,16,21,24);(1,3,5,4)(2,6,7,8)(9,11,14,22)(10,16,12,19)(13,21,15,23)(17,24,18,20);(1,4,5,3)(2,6,7,8)(9,12,14,10)(11,20,22,24)(13,18,15,17)(16,23,19,21);(2,7)(6,8)(9,13,14,15)(10,17,12,18)(11,21,22,23)(16,24,19,20)Polycyclic presentation:
2^8|a^2=fh,b^2=e,c^2=e,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,f^c=fh,g^c=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24625, #6453180, #6453182, … |
| maximal normal subgroup of | C2×#24625, #6453180, #6453182, … |
| maximal quotient of | #275232, #275237, #275244, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17613);