Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60756 |
| 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 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1091 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)(3,7)(4,6)(5,8)(9,10,14,18)(11,15,20,24)(12,17,13,22)(16,21,23,19);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,21)(13,19)(14,20)(16,22)(17,23)(18,24);(3,8)(5,7)(9,11)(10,16)(12,19)(13,21)(14,20)(15,22)(17,24)(18,23);(2,6)(5,7);(9,12,14,13)(10,17,18,22)(11,19,20,21)(15,16,24,23)Polycyclic presentation:
2^8|a^2=h,e^2=h,g^2=h,b^a=bf,c^a=cg,d^a=df,g^a=gh,c^b=cf,e^b=eh,g^c=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | #249960, #249964, #249968, … |
| maximal normal subgroup of | #249960, #249964, #249968, … |
| maximal quotient of | #6374171, #6374176, #6374887, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53751);