Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2846 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | C2×#127 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
615 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,4)(6,8)(9,10,12,11,13,14,16,15);(1,2,3,4);(5,6,7,8);(9,11,12,14,13,15,16,10)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=f,d^2=g,g^2=h,b^a=be,c^a=cf,d^a=dh88 conjugacy classes — view the full character table.
| Fitting subgroup of | #21681⋊C3 |
| maximal subgroup of | #96870, #96874, #96878, … |
| maximal normal subgroup of | #96870, #96874, #96878, … |
| maximal quotient of | #212804, #212805, #212806, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14669);