Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | #3483 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
537 conjugacy classes of subgroups; only the 189 normal subgroups are shown.
This group has 537 conjugacy classes of subgroups — too many to draw. Only its 189 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,8,6,3,4,7)(9,10)(11,20)(12,17)(13,18)(14,23)(15,19)(16,21)(22,24);(1,3,6,2)(4,8,5,7)(9,11,15,12)(10,16,19,24)(13,21,14,22)(17,23,20,18);(1,4,6,5)(2,7,3,8)(9,12,15,11)(10,17,19,20)(13,21,14,22)(16,23,24,18);(1,4,6,5)(2,7,3,8)(9,13,15,14)(10,18,19,23)(11,21,12,22)(16,17,24,20)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=g,d^2=g,e^2=g,f^2=h,b^a=be,e^a=eg,e^b=eg,d^c=dgh70 conjugacy classes — view the full character table.
| Fitting subgroup of | #17520⋊C3 |
| maximal subgroup of | #221107, #221109, #221111, … |
| maximal normal subgroup of | #221107, #221109, #221111, … |
| maximal quotient of | #164164, #164165, #164166, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10508);