Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #127 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #392 |
| prime radical | C2×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
687 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 687 conjugacy classes of subgroups — too many to draw. Only its 127 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) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6,10,13,11,15,9,14)(7,16,17,8,18,19,12,20);(2,4)(5,7)(6,8)(9,12)(10,17)(11,18)(13,19)(14,16)(15,20);(5,8)(6,12)(7,13)(9,16)(10,19)(11,20)(14,18)(15,17)Polycyclic presentation:
2^8|a^2=g,d^2=h,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=dh,f^b=fh,e^c=eh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #93904, #137392, #137393, … |
| maximal normal subgroup of | #93904, #137392, #137393, … |
| maximal quotient of | #111225, #111275, #111276, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4029);