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) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
487 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,10,15)(7,16,20,12)(8,21,23,30)(9,11,25,14)(13,24,29,22)(17,26,19,28)(18,33,34,36)(27,32,31,35);(1,3,4,2)(6,11)(7,17)(9,25)(13,21)(14,15)(18,32)(19,20)(22,24)(26,28)(29,30)(33,36)(34,35);(2,3)(5,7,10,20)(6,12,15,16)(8,18,23,34)(9,19,25,17)(11,26,14,28)(13,27,29,31)(21,36,30,33)(22,32,24,35);(5,8)(6,13)(7,18)(9,24)(10,23)(11,21)(12,27)(14,30)(15,29)(16,31)(17,32)(19,35)(20,34)(22,25)(26,36)(28,33)Polycyclic presentation:
2^8|a^2=f,b^2=fh,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×#25186, #6438852, #6438854, … |
| maximal normal subgroup of | C2×#25186, #6438852, #6438854, … |
| maximal quotient of | #282575, #282578, #282579, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18174);