Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | Aut(G) order 49152 |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C42 |
| commutator | C4 | C43 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #376 |
| prime radical | C24 | #41 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
421 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)(7,17,20,27)(8,13)(9,14)(10,16)(11,15)(12,19,26,21)(18,31,33,36)(22,28)(23,30)(24,29)(25,32,35,34);(2,4)(5,7)(6,12,15,26)(8,18)(9,19)(10,21)(11,20)(13,25,29,35)(14,17,16,27)(22,32)(23,34)(24,33)(28,31,30,36);(5,8,10,23,11,24,9,22)(6,13,16,30,15,29,14,28)(7,18,21,34,20,33,19,32)(12,25,27,36,26,35,17,31)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=g,d^2=h,g^2=h,b^a=bd,f^a=fh,e^b=eh88 conjugacy classes — view the full character table.
| Fitting subgroup of | #10699⋊C3 |
| maximal subgroup of | #93417, #128773, #128832, … |
| maximal normal subgroup of | #93417, #128773, #128832, … |
| maximal quotient of | #105257, #105258, #105259, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3687);