Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
971 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,16)(8,13)(9,14)(10,23)(11,15)(12,18)(17,32)(19,26)(20,29)(21,31)(22,30)(24,27)(25,35)(28,33)(34,36);(5,7)(6,12)(8,17)(9,18)(10,24)(11,19)(13,28)(14,26)(15,27)(16,23)(20,33)(21,35)(22,34)(25,30)(29,36)(31,32);(2,4)(5,8,11,22)(6,13,15,30)(7,17,19,34)(9,21,10,20)(12,25,27,28)(14,31,23,29)(16,32,26,36)(18,35,24,33);(5,9,11,10)(6,14,15,23)(7,18,19,24)(8,20,22,21)(12,26,27,16)(13,29,30,31)(17,33,34,35)(25,32,28,36)Polycyclic presentation:
2^8|c^2=h,d^2=h,e^2=h,b^a=be,c^a=cf,e^a=eh,g^a=gh,c^b=cg,e^b=eh,d^c=dh,e^c=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #376208, #381473, #388380, … |
| maximal normal subgroup of | #376208, #381473, #388380, … |
| maximal quotient of | #233671, #233675, #233679, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16391);