Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #459 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
147 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(5,6)(7,17)(8,14)(9,24)(10,27)(11,16)(12,18)(13,22)(15,26)(19,32)(20,34)(21,28)(23,33)(25,29)(30,35)(31,36);(1,3,4,2)(5,7)(6,12)(8,18)(9,25)(10,19)(11,21)(13,28)(14,32)(15,29)(16,31)(17,24)(20,26)(22,35)(23,36)(27,30)(33,34);(5,8,10,22,11,23,26,9)(6,13,15,14,16,24,27,33)(7,18,19,35,21,36,20,25)(12,28,29,32,31,17,30,34)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=e,d^2=gh,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,d^b=dg,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #136388, #136390, #136509, … |
| maximal normal subgroup of | #136388, #136390, #136509, … |
| maximal quotient of | #140380, #140401, #140417, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6168);