Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
939 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 939 conjugacy classes of subgroups — too many to draw. Only its 199 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) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(9,10,12,11)(13,14,17,16)(15,19,20,18);(1,2,4,3)(6,8)(10,11)(15,20)(18,19);(1,2,4,3)(5,6,7,8)(9,11,12,10)(13,15)(14,18)(16,19)(17,20);(1,3,4,2)(13,16,17,14)(15,19,20,18)Polycyclic presentation:
2^8|a^2=ef,b^2=h,c^2=fg,d^2=f,b^a=be,c^a=cf,d^a=dh,c^b=cg64 conjugacy classes — view the full character table.
| maximal subgroup of | #101128, #101144, #103313, … |
| maximal normal subgroup of | #101128, #101144, #103313, … |
| maximal quotient of | #154159, #154268, #154305, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8072);