Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10792 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C4×C22⋊C4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1434 |
| prime radical | C24 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,12,16,27)(4,17)(5,10)(6,20)(7,11)(8,14,23,15)(9,18)(13,22,29,32)(19,24)(21,25)(26,30,31,28);(1,3)(2,8,11,23)(4,13)(5,14)(6,15)(7,16)(9,22,25,32)(10,12,20,27)(17,31,24,26)(18,28)(19,29)(21,30);(1,4,5,18,7,19,6,21)(2,9,10,24,11,25,20,17)(3,13,14,28,16,29,15,30)(8,22,12,26,23,32,27,31)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=d,d^2=h,e^2=h,b^a=bd,c^a=ce,e^a=eh,g^a=gh,f^b=fh,e^c=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10810, #353608, #356526, … |
| maximal normal subgroup of | C2×#10810, #353608, #356526, … |
| maximal quotient of | #108430, #108431, #108434, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3798);