Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
473 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 473 conjugacy classes of subgroups — too many to draw. Only its 167 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) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6,10,15,12,16,11,20)(7,9,14,19,17,18,13,8);(1,3,4,2)(5,7,12,17)(6,8,16,19)(9,15,18,20)(10,13,11,14);(5,8,11,18,12,19,10,9)(6,13,15,7,16,14,20,17);(5,9,10,19,12,18,11,8)(6,14,15,17,16,13,20,7)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=f,d^2=e,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,d^b=de,e^b=eh,g^b=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #136491, #136510, #136535, … |
| maximal normal subgroup of | #136491, #136510, #136535, … |
| maximal quotient of | #231466, #231473, #231485, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16186);