Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| socle | C22 | C23×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1361 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,16)(8,13)(9,14)(10,24)(11,15)(12,20)(17,29)(18,30)(19,26)(21,27)(22,28)(23,34)(25,32)(31,35)(33,36);(5,7,10,20,11,18,9,19)(6,12,14,16,15,26,24,30)(8,17,23,32,21,33,22,31)(13,25,28,29,27,35,34,36);(2,4)(5,8)(6,13)(7,17)(9,22)(10,23)(11,21)(12,25)(14,28)(15,27)(16,29)(18,33)(19,31)(20,32)(24,34)(26,35)(30,36);(7,18)(8,21)(12,26)(13,27)(16,30)(19,20)(22,23)(28,34);(5,9,11,10)(6,14,15,24)(7,19,18,20)(8,22,21,23)(12,16,26,30)(13,28,27,34)(17,31,33,32)(25,29,35,36)Polycyclic presentation:
2^8|b^2=f,e^2=h,f^2=h,b^a=bf,c^a=cg,f^a=fh,d^b=dh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #129727, #129728, #129775, … |
| maximal normal subgroup of | #129727, #129728, #129775, … |
| maximal quotient of | #6353731, #6353734, #6353739, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53441);