Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| 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 | #386 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1436 |
| prime radical | C24 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
336 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,10,7,11,5,20)(3,12,15,24,16,23,14,8)(4,17,18,9,19,25,21,26)(13,28,29,32,31,22,30,27);(1,3)(2,8,11,24)(4,13)(5,14)(6,15)(7,16)(9,22,26,28)(10,12,20,23)(17,32,25,27)(18,29)(19,31)(21,30);(1,4,6,18,7,19,5,21)(2,9,10,25,11,26,20,17)(3,13,15,29,16,31,14,30)(8,22,12,27,24,28,23,32)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=f,d^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,e^b=eh,d^c=dh,e^c=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10905, #354322, #355278, … |
| maximal normal subgroup of | C2×#10905, #354322, #355278, … |
| maximal quotient of | #109536, #109537, #109538, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3893);