Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | Aut(G) order 32768 |
| 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 | C24 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
441 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)(5,6,10,14,11,15,9,24)(7,12,19,27,17,26,18,33)(8,20,22,28,23,34,21,13)(16,29,32,35,30,36,31,25);(2,4)(5,7,10,19,11,17,9,18)(6,12,14,27,15,26,24,33)(8,16,22,32,23,30,21,31)(13,25,20,29,28,35,34,36);(5,8,9,21,11,23,10,22)(6,13,14,20,15,28,24,34)(7,16,18,31,17,30,19,32)(12,25,27,29,26,35,33,36);(7,17)(12,26)(16,30)(18,19)(25,35)(27,33)(29,36)(31,32)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=f,f^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #124286, #140638, #147182, … |
| maximal normal subgroup of | #124286, #140638, #147182, … |
| maximal quotient of | #203502, #203503, #203504, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14151);