Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8304 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator, prime radical | C24 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
320 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 320 conjugacy classes of subgroups — too many to draw. Only its 85 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 — 3 generators on 20 points:
(2,4)(5,6)(7,9,11,12)(8,10)(13,14,18,19,17,15,16,20);(5,7,8,11)(6,9)(10,12)(13,15,18,20,17,14,16,19);(1,2,3,4)(6,10)(9,12)(13,16,17,18)(14,19,15,20)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=eg,f^2=h,b^a=bd,c^a=ce,d^a=dg,c^b=ch,d^b=dg,f^b=fg52 conjugacy classes — view the full character table.
| maximal subgroup of | #97737, #97739, #97745, … |
| maximal normal subgroup of | #97737, #97739, #97745, … |
| maximal quotient of | #101974, #101976, #101978, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2642);