Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #386 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
589 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,9,15,11,16,10,14)(7,17,20,28,21,27,19,12)(8,22,24,29,25,30,23,13)(18,26,32,35,33,36,34,31);(5,7,10,19,11,21,9,20)(6,12,14,27,16,28,15,17)(8,18,24,32,25,33,23,34)(13,26,22,35,29,36,30,31);(2,4)(5,8)(6,13)(7,18)(9,23)(10,24)(11,25)(12,26)(14,22)(15,30)(16,29)(17,31)(19,32)(20,34)(21,33)(27,35)(28,36)Polycyclic presentation:
2^8|a^2=f,b^2=g,d^2=h,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,d^b=dh,e^c=eh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10782, #353462, C4×#1881, … |
| maximal normal subgroup of | C2×#10782, #353462, C4×#1881, … |
| maximal quotient of | #105867, #105870, #105874, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3770);