Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
439 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)(3,6)(4,5)(7,8)(9,10)(11,20)(12,23)(13,22)(14,18)(15,19)(16,17)(21,24);(2,5)(6,8)(9,11,14,17,15,22,23,21)(10,16,12,20,19,24,18,13);(1,3,4,7)(2,6,5,8)(9,12,23,10,15,18,14,19)(11,16,21,13,22,24,17,20);(2,5)(6,8)(9,13,14,16,15,20,23,24)(10,17,12,22,19,21,18,11)Polycyclic presentation:
2^8|b^2=f,c^2=eh,d^2=f,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,g^b=gh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#27288, #6440834, #6440840, … |
| maximal normal subgroup of | C2×#27288, #6440834, #6440840, … |
| maximal quotient of | #313581, #313584, #313585, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20276);