Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C22×C4⋊C4 |
| prime radical | C23 | C22×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
321 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(7,8)(9,10,13,16,14,17,12,22)(11,18,20,23,21,24,19,15);(1,2,5,6)(3,7,4,8);(1,3,5,4)(2,7,6,8)(9,11,12,19,14,21,13,20)(10,15,16,18,17,23,22,24);(1,4,5,3)(2,7,6,8)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=ef,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,d^b=de70 conjugacy classes — view the full character table.
| Fitting subgroup of | #22746⋊C3 |
| maximal subgroup of | #138462, #138463, #138472, … |
| maximal normal subgroup of | #138462, #138463, #138472, … |
| maximal quotient of | #226424, #226425, #226426, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15734);