Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2433 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
880 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 880 conjugacy classes of subgroups — too many to draw. Only its 175 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,4)(10,14)(11,16)(12,15);(5,6)(7,8)(9,10,12,14,13,11,15,16);(1,2,3,4)(6,8);(9,11,12,16,13,10,15,14)Polycyclic presentation:
2^8|b^2=e,c^2=f,d^2=e,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg58 conjugacy classes — view the full character table.
| maximal subgroup of | #301861, #301862, #301917, … |
| maximal normal subgroup of | #301861, #301862, #301917, … |
| maximal quotient of | #298603, #298609, #298616, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18762);