Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1366 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #333 |
| commutator | C42 | C42 |
| Frattini, cosocle | C2×C4×C8 | C22 |
| prime radical | C23 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
111 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2,3,4)(5,6)(7,15,17,28)(8,20)(9,12)(10,24)(11,16,25,19)(13,23)(14,21)(18,26,32,36)(22,29)(27,34)(30,33,35,31);(5,7,8,18,21,19,34,33,9,17,22,32,10,16,23,31)(6,11,13,26,24,15,29,30,12,25,27,36,14,28,20,35)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fh,e^2=fg,g^2=h,b^a=bc,c^a=cf,e^a=efgh,g^a=gh,c^b=cg,d^b=dh,f^b=fh,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7271, #127700, #127745, … |
| maximal normal subgroup of | C2×#7271, #127700, #127745, … |
| maximal quotient of | #93083, #93084, #93085, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 259);