Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61049 |
| 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 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #525 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1053 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10,15,11)(12,20,22,24)(13,17,14,19)(16,23,18,21);(1,3)(2,5)(4,8)(6,7)(9,11,15,10)(12,18,22,16)(13,17,14,19)(20,21,24,23);(9,12)(10,16)(11,18)(13,21)(14,23)(15,22)(17,24)(19,20);(3,8)(5,7);(9,13,15,14)(10,17,11,19)(12,21,22,23)(16,24,18,20)Polycyclic presentation:
2^8|a^2=fh,b^2=h,e^2=h,g^2=h,b^a=bf,c^a=cg,g^a=gh,d^b=df,e^b=eh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #220536, #220539, #220555, … |
| maximal normal subgroup of | #220536, #220539, #220555, … |
| maximal quotient of | #6379073, #6379303, #6379700, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54162);