Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32878 |
| automorphism | Aut(G) order 393216 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
537 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(6,8)(9,10,13,16)(11,17,19,24)(12,20,22,15)(14,21,23,18);(1,2,3,4);(5,6,7,8);(9,11,13,19)(10,14,16,23)(12,21,22,18)(15,24,20,17)Polycyclic presentation:
2^8|a^2=h,b^2=e,c^2=f,d^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,g^a=gh,g^d=gh58 conjugacy classes — view the full character table.
| Fitting subgroup of | #33163⋊C3 |
| maximal subgroup of | #100328, #100329, #100360, … |
| maximal normal subgroup of | #100328, #100329, #100360, … |
| maximal quotient of | #351647, #351649, #351651, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26151);