Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1418 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #357 |
| commutator | C16 | C42 |
| Frattini, cosocle | C22×C16 | C22 |
| prime radical | C23 | D16 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
89 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)(5,6,9,15)(7,19,21,27)(8,24,26,16)(10,14,28,23)(11,22,29,36)(12,20,13,18)(17,32,34,25)(30,35,31,33);(5,7,10,22,11,23,19,6,13,16,31,17,32,33,26,20,9,21,28,36,29,14,27,15,12,24,30,34,25,35,8,18)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fg,e^2=f,f^2=gh,g^2=h,b^a=bc,c^a=cf,e^a=efgh,f^a=fg,g^a=gh,c^b=ch52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7477, #151776, C4×#1418, … |
| maximal normal subgroup of | C2×#7477, #151776, C4×#1418, … |
| maximal quotient of | #94025, #94048, #94051, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 465);