Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#578 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
651 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)(5,6,11,15)(7,17,20,27)(8,21,25,13)(9,19,12,28)(10,16,29,14)(18,34,36,31)(22,30,35,23)(24,32,26,33);(5,7,11,20)(6,12,15,9)(8,22,25,35)(10,28,29,19)(13,31,21,34)(14,27,16,17)(18,24,36,26)(23,32,30,33);(2,4)(5,8,11,25)(6,13,15,21)(7,18,20,36)(9,23,12,30)(10,24,29,26)(14,32,16,33)(17,31,27,34)(19,35,28,22);(5,9)(6,7)(8,23)(10,17)(11,12)(13,18)(14,28)(15,20)(16,19)(21,36)(22,32)(24,31)(25,30)(26,34)(27,29)(33,35)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,g^b=gh,g^c=gh,e^d=eh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24462, #6437902, #6437906, … |
| maximal normal subgroup of | C2×#24462, #6437902, #6437906, … |
| maximal quotient of | #268633, #268638, #268649, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17450);