Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #386 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
495 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 495 conjugacy classes of subgroups — too many to draw. Only its 143 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 — 3 generators on 20 points:
(2,4)(5,6,11,14)(7,10,13,9)(8,15)(12,17)(16,20)(18,19);(5,7,10,6,11,13,9,14)(8,15,17,12,18,19,16,20);(1,2,3,4)(5,8,9,16,11,18,10,17)(6,12,7,15,14,20,13,19)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=ef,d^2=h,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,d^b=dh,e^c=eh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #94185, #94187, #94189, … |
| maximal normal subgroup of | #94185, #94187, #94189, … |
| maximal quotient of | #110066, #110067, #110070, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3947);