Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C23 | C2×C22⋊C4 |
| prime radical | C22×#41 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
560 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 560 conjugacy classes of subgroups — too many to draw. Only its 125 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:
(1,2)(3,4)(6,10)(7,8)(11,12)(13,14,15,17,16,18,19,20);(5,6)(7,10)(8,12)(9,11);(2,4)(5,7,9,8)(6,10,11,12)Polycyclic presentation:
2^8|a^2=f,c^2=g,d^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,d^b=dg76 conjugacy classes — view the full character table.
| maximal subgroup of | #97678, #97680, #97681, … |
| maximal normal subgroup of | #97678, #97680, #97681, … |
| maximal quotient of | #96912, #96914, #96915, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1300);