Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | Aut(G) = #72310497 (order 2048) · Inn(G) = C2×C42 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×#97 |
| prime radical | C24 | M16 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
210 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6,9,15,10,12,18,19,11,7,13,16,8,14,17,20);(2,4)(6,12,7,14)(9,13)(15,20,16,19)(17,18);(5,7,9,16,10,14,18,20,11,6,13,15,8,12,17,19)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=e,d^2=h,e^2=g,g^2=h,b^a=bd,f^a=fh,c^b=cg,e^b=eh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #128465, #128466, #128467, … |
| maximal normal subgroup of | #128465, #128466, #128467, … |
| maximal quotient of | #128255, #128256, #128267, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5604);