Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60756 |
| 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 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1054 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)(10,15)(11,19)(12,14)(16,23)(20,24)(21,22);(1,3,4,8)(2,5,6,7)(9,10,13,15)(11,20,21,16)(12,17,14,18)(19,23,22,24);(3,8)(5,7)(9,11)(10,16)(12,19)(13,21)(14,22)(15,20)(17,24)(18,23);(2,6)(5,7)(9,12,13,14)(10,17,15,18)(11,19,21,22)(16,24,20,23);(10,15)(16,20)(17,18)(23,24)Polycyclic presentation:
2^8|b^2=f,d^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 | #6668400, #6668401, #6668412, … |
| maximal normal subgroup of | #6668400, #6668401, #6668412, … |
| maximal quotient of | #6374286, #6374546, #6375011, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53803);