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 | #3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
977 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)(7,15)(8,12)(9,21)(10,14)(11,17)(13,24)(16,32)(18,30)(19,28)(20,31)(22,26)(23,29)(25,35)(27,34)(33,36);(5,7,10,18)(6,11,14,26)(8,16,19,33)(9,22,24,17)(12,27,28,25)(13,30,21,15)(20,35,23,34)(29,36,31,32);(2,4)(5,8)(6,12)(7,16)(9,23)(10,19)(11,25)(13,31)(14,28)(15,32)(17,35)(18,33)(20,24)(21,29)(22,34)(26,27)(30,36);(8,19)(12,28)(16,33)(20,23)(25,27)(29,31)(32,36)(34,35)Polycyclic presentation:
2^8|b^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 | #376206, #381471, #388382, … |
| maximal normal subgroup of | #376206, #381471, #388382, … |
| maximal quotient of | #233669, #233672, #233677, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16389);