Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
427 conjugacy classes of subgroups; only the 211 normal subgroups are shown.
This group has 427 conjugacy classes of subgroups — too many to draw. Only its 211 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) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6)(7,12)(8,10)(9,11)(14,17)(15,18)(19,20);(5,7,9,8)(6,10,11,12);(13,14,18,20,16,19,15,17);(1,3,4,2)(5,8,9,7)(6,10,11,12)(14,19)(17,20)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=fh,d^2=eg,f^2=h,b^a=be,c^a=cf,f^a=fh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #100093, #100103, #124642, … |
| maximal normal subgroup of | #100093, #100103, #124642, … |
| maximal quotient of | #213368, #213378, #213389, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14715);