Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| 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 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
671 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)(7,16,18,12,19,28,24,27)(8,20,21,13,22,30,25,29)(17,26,33,36,34,35,32,31);(2,4)(5,7,10,18,11,19,9,24)(6,12,15,28,14,27,23,16)(8,17,21,33,22,34,25,32)(13,26,30,36,29,35,20,31);(5,8)(6,13)(7,17)(9,25)(10,21)(11,22)(12,26)(14,29)(15,30)(16,31)(18,33)(19,34)(20,23)(24,32)(27,35)(28,36);(6,14)(12,27)(13,29)(15,23)(16,28)(20,30)(26,35)(31,36)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,e^b=eh,f^b=fh,e^c=eh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #137718, #365167, #365168, … |
| maximal normal subgroup of | #137718, #365167, #365168, … |
| maximal quotient of | #191863, #191864, #191865, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13872);