Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #380 |
| prime radical | C24 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
365 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,14,11,15,9,25)(7,16,18,12,20,27,19,28)(8,21,23,29,24,35,22,13)(17,26,31,34,33,36,32,30);(1,2,3,4)(5,7)(6,12)(8,17)(9,19)(10,18)(11,20)(13,26)(14,27)(15,28)(16,25)(21,34)(22,32)(23,31)(24,33)(29,36)(30,35);(5,8,9,22,11,24,10,23)(6,13,14,21,15,29,25,35)(7,17,19,32,20,33,18,31)(12,26,27,34,28,36,16,30)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=e,d^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #146754, #146762, #146766, … |
| maximal normal subgroup of | #146754, #146762, #146766, … |
| maximal quotient of | #108900, #108901, #108902, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3860);