Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | #3050 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
453 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,14,11,15,9,25)(7,16,19,27,20,32,18,12)(8,13,24,30,22,29,23,36)(17,31,34,28,21,35,33,26);(5,7,9,18,11,20,10,19)(6,12,14,16,15,27,25,32)(8,21,23,34,22,17,24,33)(13,28,30,35,29,26,36,31);(2,4)(8,22)(13,29)(17,21)(23,24)(26,28)(30,36)(31,35)(33,34);(5,8,10,24,11,22,9,23)(6,13,14,30,15,29,25,36)(7,17,19,34,20,21,18,33)(12,26,16,31,27,28,32,35)Polycyclic presentation:
2^8|a^2=g,b^2=e,d^2=g,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,d^b=dh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #124785, #134477, #134507, … |
| maximal normal subgroup of | #124785, #134477, #134507, … |
| maximal quotient of | #210731, #210732, #210733, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14535);