Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
663 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)(5,6,10,15,11,14,9,23)(7,16,19,12,18,27,24,28)(8,20,21,13,22,30,25,29)(17,26,34,35,32,36,33,31);(2,4)(5,7,10,19,11,18,9,24)(6,12,15,27,14,28,23,16)(8,17,21,34,22,32,25,33)(13,26,30,35,29,36,20,31);(5,8)(6,13)(7,17)(9,25)(10,21)(11,22)(12,26)(14,29)(15,30)(16,31)(18,32)(19,34)(20,23)(24,33)(27,35)(28,36);(6,14)(7,18)(13,29)(15,23)(17,32)(19,24)(20,30)(33,34)Polycyclic presentation:
2^8|a^2=g,b^2=g,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,d^b=dh,e^b=eh,f^b=fh,e^c=eh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #137749, #360558, #360559, … |
| maximal normal subgroup of | #137749, #360558, #360559, … |
| maximal quotient of | #191869, #191870, #191871, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13874);