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 | #3482 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
510 conjugacy classes of subgroups; only the 189 normal subgroups are shown.
This group has 510 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)(9,10,15,12)(11,18,21,24)(13,17,14,22)(16,23,19,20);(1,3)(2,6)(4,8)(5,7)(9,11)(10,16)(12,19)(13,20)(14,23)(15,21)(17,24)(18,22);(1,4,6,5)(2,7,3,8)(9,12,15,10)(11,19,21,16)(13,17,14,22)(18,20,24,23);(1,4,6,5)(2,7,3,8)(9,13,15,14)(10,17,12,22)(11,20,21,23)(16,24,19,18)Polycyclic presentation:
2^8|a^2=f,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 | #17518⋊C3 |
| maximal subgroup of | #221105, #221108, #221112, … |
| maximal normal subgroup of | #221105, #221108, #221112, … |
| maximal quotient of | #164134, #164135, #164136, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10506);