Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C23 | C2×C22⋊C4 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
281 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 281 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,4,3)(6,10)(7,8)(11,12)(13,14,17,20)(15,21,22,28)(16,24,25,19)(18,26,27,23);(5,6,8,11,9,12,7,10)(13,15)(14,18,20,27)(16,23)(17,22)(19,21,24,28)(25,26);(1,3,4,2)(5,7,9,8)(6,11,12,10)(15,22)(18,27)(21,28)(23,26)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=eg,d^2=g,f^2=h,b^a=bd,c^a=ch,d^a=dg,f^a=fg,c^b=cg64 conjugacy classes — view the full character table.
| maximal subgroup of | #97878, #97881, #97883, … |
| maximal normal subgroup of | #97878, #97881, #97883, … |
| maximal quotient of | #96711, #96715, #96749, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1167);