Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3186 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#466 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
531 conjugacy classes of subgroups; only the 171 normal subgroups are shown.
This group has 531 conjugacy classes of subgroups — too many to draw. Only its 171 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:
(1,2,4,3)(5,6)(7,11)(8,10)(9,12)(14,17)(15,19)(18,20);(1,3,4,2)(5,7)(6,9)(8,12)(10,11)(13,14,16,17)(15,19,20,18);(7,12)(9,11)(13,15,17,19,16,20,14,18);(6,10)(9,11)(13,14,16,17)(15,18,20,19)Polycyclic presentation:
2^8|a^2=g,b^2=gh,c^2=f,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce64 conjugacy classes — view the full character table.
| maximal subgroup of | #136226, #136274, #137387, … |
| maximal normal subgroup of | #136226, #136274, #137387, … |
| maximal quotient of | #218744, #218756, #218792, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15172);