Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| socle | C22 | C23×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
980 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(6,8)(9,10)(11,15)(12,19)(13,21)(14,17)(16,18)(20,23)(22,24);(1,2)(3,6)(4,5)(7,8)(9,11,14,13)(10,15,17,21)(12,18,20,22)(16,23,24,19);(9,12,14,20)(10,16,17,24)(11,18,13,22)(15,23,21,19);(2,5)(6,8)(9,11,14,13)(10,15,17,21)(12,18,20,22)(16,23,24,19);(1,3,4,7)(2,6,5,8)(9,11,14,13)(10,15,17,21)(12,18,20,22)(16,23,24,19)Polycyclic presentation:
2^8|b^2=h,c^2=h,d^2=h,e^2=fh,g^2=h,b^a=bf,c^a=cg,g^a=gh,d^b=df,g^c=gh70 conjugacy classes — view the full character table.
| Fitting subgroup of | #61436⋊C3 |
| maximal subgroup of | #131789, #131813, #221109, … |
| maximal normal subgroup of | #131789, #131813, #221109, … |
| maximal quotient of | #6379540, #6380375, #6380883, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54424);