Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11651 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#102 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
167 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10,14,19,22,36,28,18,13,17,29,35,15,20,12,27)(11,21,25,16,31,33,23,38,24,39,40,32,26,37,30,34);(2,5)(6,7)(9,11,22,26,13,24,15,31)(10,16,20,38,17,32,36,34)(12,25,14,30,28,40,29,23)(18,37,19,21,27,33,35,39);(1,3,4,8)(2,6,5,7)(10,17)(16,32)(18,27)(19,35)(20,36)(21,39)(33,37)(34,38)Polycyclic presentation:
2^8|a^2=f,b^2=defg,c^2=eh,d^2=g,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,d^b=dg,f^b=fg,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #131700, #131701, #131709, … |
| maximal normal subgroup of | #131700, #131701, #131709, … |
| maximal quotient of | #141860, #141876, #142098, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6268);