Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60460 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | #3500 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1033 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,17)(16,18)(19,22)(20,23)(21,24);(2,5)(6,8)(9,11,15,18,16,12,14,10)(13,19,23,24,21,20,22,17);(1,3,4,7)(2,6,5,8)(9,12,14,18,16,11,15,10)(13,20,22,24,21,19,23,17);(9,13)(10,17)(11,19)(12,20)(14,22)(15,23)(16,21)(18,24);(13,21)(17,24)(19,20)(22,23)Polycyclic presentation:
2^8|b^2=g,c^2=f,f^2=h,g^2=h,b^a=bf,c^a=cg,f^a=fh,g^a=gh,c^b=ch,e^d=eh58 conjugacy classes — view the full character table.
| Fitting subgroup of | #60625⋊C3 |
| maximal subgroup of | #249819, #249820, #249821, … |
| maximal normal subgroup of | #249819, #249820, #249821, … |
| maximal quotient of | #6371057, #6371700, #6371701, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53613);