Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | Aut(G) order 8192 |
| 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 | C4×D8 |
| prime radical | C23 | #135 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
155 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,15,11,16,9,14,12,17,26,33,25,32,8,24)(7,18,21,13,22,29,27,30,23,34,19,31,28,36,20,35);(1,2,3,4)(5,7)(6,13)(8,19)(9,21)(10,27)(11,28)(12,23)(14,29)(15,34)(16,35)(17,31)(18,33)(20,26)(22,25)(24,36)(30,32);(5,8,25,26,12,9,11,10)(6,14,32,15,17,24,16,33)(7,19,22,20,23,21,28,27)(13,29,30,34,31,36,35,18)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=deg,d^2=gh,e^2=g,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 | #94056, #94057, #129164, … |
| maximal normal subgroup of | #94056, #94057, #129164, … |
| maximal quotient of | #129096, #129111, #129128, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5659);