Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2846 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C24 |
| commutator | C23 | C23×C4 |
| prime radical | C26 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3175 conjugacy classes of subgroups; only the 431 normal subgroups are shown.
This group has 3175 conjugacy classes of subgroups — too many to draw. Only its 431 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 — 4 generators on 20 points:
(2,4)(5,6,7,8)(10,12)(14,17)(18,20);(13,14)(15,18)(16,17)(19,20);(9,10)(11,12)(13,15,16,19)(14,18,17,20);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,18)(16,17)(19,20)Polycyclic presentation:
2^8|a^2=h,c^2=e,b^a=be,c^a=cf,d^a=dg88 conjugacy classes — view the full character table.
| maximal subgroup of | #95130, #95138, #95196, … |
| maximal normal subgroup of | #95130, #95138, #95196, … |
| maximal quotient of | #154077, #154078, #154082, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7669);