Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×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, prime radical | C23 | C2×C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
249 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 249 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) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,4,3)(5,6,10,12)(7,9,11,8)(13,14)(15,21)(16,24)(17,20)(18,23)(19,25)(22,27)(26,28);(5,7,9,12,10,11,8,6)(13,15)(14,18,20,26)(16,23)(17,22)(19,21,24,27)(25,28);(1,3,4,2)(5,8,10,9)(6,11,12,7)(15,22)(18,26)(21,27)(23,28)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=e,d^2=g,f^2=h,b^a=bd,c^a=ch,d^a=dg,f^a=fg,c^b=cgh76 conjugacy classes — view the full character table.
| maximal subgroup of | #97991, #97992, #97994, … |
| maximal normal subgroup of | #97991, #97992, #97994, … |
| maximal quotient of | #96718, #96722, #96745, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1170);