Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D8 |
| commutator | C8 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×#107 |
| prime radical | C23 | C2×Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
153 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(5,6,8,13,21,28,35,36,10,12,23,27,9,14,22,30)(7,15,16,25,17,32,33,24,19,31,34,26,18,29,20,11);(5,7,9,17,10,19,21,18)(6,11,14,25,12,24,28,26)(8,20,22,16,23,33,35,34)(13,29,30,15,27,32,36,31);(1,3,4,2)(6,12)(11,24)(13,27)(14,28)(15,31)(25,26)(29,32)(30,36)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=deg,d^2=gh,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ch,d^b=dg,e^b=eg,g^b=gh64 conjugacy classes — view the full character table.
| maximal subgroup of | #94058, #129166, #129178, … |
| maximal normal subgroup of | #94058, #129166, #129178, … |
| maximal quotient of | #129099, #129114, #129129, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5664);