Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8528 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #379 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | #3549 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
444 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,5,6,4,3,7,8)(10,14)(11,12)(15,16);(5,7)(6,8)(9,10)(11,16)(12,14)(13,15);(1,3)(2,4)(5,6)(7,8)(9,10)(11,14)(12,16)(13,15)Polycyclic presentation:
2^8|a^2=f,d^2=h,e^2=g,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,e^b=egh,f^b=fgh,d^c=dgh,e^c=eg,f^c=fgh40 conjugacy classes — view the full character table.
| Fitting subgroup of | #8530⋊C3, #72340797 |
| maximal subgroup of | #152567, #152570, #153133, … |
| maximal normal subgroup of | #152567, #152570, #153133, … |
| maximal quotient of | #97044, #97048, #97260, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1518);