Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D4 |
| commutator | C4 | C2×C4×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×C4⋊C8 |
| prime radical | C23 | C2×M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
179 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)(7,19,22,32)(8,24)(9,15)(10,16)(11,17)(12,18)(13,21,29,27)(14,26)(20,30,34,33)(23,28,35,36)(25,31);(5,7,10,23,11,21,26,34,12,22,25,35,9,27,8,20)(6,13,16,30,17,19,14,28,18,29,31,33,15,32,24,36);(1,3,4,2)(5,8,9,25,12,26,11,10)(6,14,15,16,18,24,17,31)(7,20,27,35,22,34,21,23)(13,28,32,30,29,36,19,33)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=de,d^2=h,f^2=g,g^2=h,b^a=bd,c^a=ch,d^b=dh,e^b=eh88 conjugacy classes — view the full character table.
| maximal subgroup of | #124410, #124411, #125269, … |
| maximal normal subgroup of | #124410, #124411, #125269, … |
| maximal quotient of | #129039, #129048, #129067, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5644);