Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13357 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C23 |
| prime radical | #2733 | C2 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
256 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 256 conjugacy classes of subgroups — too many to draw. Only its 45 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:
(2,3)(6,10)(7,13)(8,16)(11,19)(12,14)(17,20);(1,2)(3,4)(6,11)(8,16)(10,19)(12,17)(14,20)(15,18);(1,3,4,2)(5,6)(7,14)(8,17)(9,12)(10,13)(11,16)(15,19)(18,20)Polycyclic presentation:
2^8|c^2=d,e^2=h,f^2=h,b^a=bd,c^a=ce,e^a=eh,f^a=fgh,c^b=cf,e^b=eg,f^b=fh,e^c=eh,f^c=fh,g^c=gh,f^e=fh31 conjugacy classes — view the full character table.
| maximal subgroup of | #151224, #151225, #151232, … |
| maximal normal subgroup of | #151224, #151225, #151232, … |
| maximal quotient of | #144666, #144702, #144735, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6414);