Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,19,22,20,21,23,24);(1,2,5,6)(3,7,4,8)(9,11,12,15)(10,13,14,16);(1,3,5,4)(2,7,6,8)(10,14)(13,16);(1,4,5,3)(2,7,6,8)(18,21)(22,24)Polycyclic presentation:
2^8|a^2=g,b^2=ef,c^2=e,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=dh,d^b=de76 conjugacy classes — view the full character table.
| maximal subgroup of | #163743, #163744, #163748, … |
| maximal normal subgroup of | #163743, #163744, #163748, … |
| maximal quotient of | #356255, #356438, #356440, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15629);