Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
865 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(6,7)(9,10,14,15)(11,13,12,16)(17,18,22,20)(19,23,21,24);(1,2,5,4)(3,7,8,6)(10,15)(11,12)(18,20)(19,21);(9,11,14,12)(10,16,15,13)(17,19,22,21)(18,23,20,24);(1,3,5,8)(2,6,4,7)(9,12,14,11)(10,13,15,16)(17,20,22,18)(19,23,21,24);(1,4,5,2)(3,7,8,6)(9,13)(10,11)(12,15)(14,16)(17,21,22,19)(18,23,20,24)Polycyclic presentation:
2^8|a^2=h,b^2=fh,c^2=h,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#57752, #8101461, #8101961, … |
| maximal normal subgroup of | C2×#57752, #8101461, #8101961, … |
| maximal quotient of | #585567, #586888, #587215, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50740);