Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61591 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | #3483 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
936 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)(3,7)(9,10,13,17)(11,18,14,15)(12,20,21,24)(16,22,23,19);(1,2)(3,6)(4,5)(7,8)(9,11,13,14)(10,15,17,18)(12,19,21,22)(16,20,23,24);(9,12,13,21)(10,16,17,23)(11,19,14,22)(15,20,18,24);(2,5)(6,8)(9,11,13,14)(10,15,17,18)(12,19,21,22)(16,20,23,24);(1,3,4,7)(2,6,5,8)(9,11,13,14)(10,15,17,18)(12,19,21,22)(16,20,23,24)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=h,d^2=h,e^2=f,g^2=h,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gh,d^b=dfh,g^c=gh52 conjugacy classes — view the full character table.
| Fitting subgroup of | #62002⋊C3 |
| maximal subgroup of | #244709, #244712, #244719, … |
| maximal normal subgroup of | #244709, #244712, #244719, … |
| maximal quotient of | #6386421, #6388454, #6388945, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54990);