Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1005 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(10,13)(11,15)(17,18,20,22)(19,23,24,21);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,21)(20,24)(22,23);(1,4,6,5)(2,7,8,3)(9,11,12,15)(10,14,13,16)(18,22)(19,24);(2,8)(3,7)(10,13)(14,16)(19,24)(21,23);(1,5,6,4)(2,3,8,7)Polycyclic presentation:
2^8|a^2=h,c^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df64 conjugacy classes — view the full character table.
| maximal subgroup of | #97519, #165779, #165786, … |
| maximal normal subgroup of | #97519, #165779, #165786, … |
| maximal quotient of | #520782, #520823, #520883, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45266);