Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C2×D42 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
758 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 758 conjugacy classes of subgroups — too many to draw. Only its 111 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)(5,6,9,11,8,7,12,10)(13,14)(15,18)(16,19)(17,20);(2,4)(9,12)(10,11)(14,18)(15,16)(19,20);(5,7)(6,8)(9,10)(11,12)(14,19)(15,16)(18,20)Polycyclic presentation:
2^8|a^2=f,d^2=h,e^2=g,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eg,f^b=fgh,d^c=dg,e^c=eg52 conjugacy classes — view the full character table.
| maximal subgroup of | #94538, #152554, #153117, … |
| maximal normal subgroup of | #94538, #152554, #153117, … |
| maximal quotient of | #97486, #97496, #97497, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1431);