Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×#121 |
| prime radical | C2×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
843 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)(5,6,10,16,11,15,9,14)(7,12,18,26,19,28,20,27)(8,21,22,13,23,29,24,30)(17,31,33,25,34,35,32,36);(5,7,10,20,11,19,9,18)(6,12,14,26,15,28,16,27)(8,17,24,33,23,34,22,32)(13,25,29,31,30,36,21,35);(2,4)(5,8)(6,13)(7,17)(9,22)(10,24)(11,23)(12,25)(14,29)(15,30)(16,21)(18,32)(19,34)(20,33)(26,31)(27,35)(28,36);(5,9,11,10)(6,14,15,16)(7,18,19,20)(8,22,23,24)(12,26,28,27)(13,29,30,21)(17,32,34,33)(25,31,36,35)Polycyclic presentation:
2^8|a^2=g,b^2=eg,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fh,g^c=gh64 conjugacy classes — view the full character table.
| maximal subgroup of | #354318, #354319, #356682, … |
| maximal normal subgroup of | #354318, #354319, #356682, … |
| maximal quotient of | #196172, #196178, #196182, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13991);