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 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
763 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,5)(4,7)(6,8)(9,10,13,15)(11,16,12,14)(17,18)(19,21)(20,27)(22,24)(23,25)(26,29)(28,31)(30,32);(2,3)(4,8)(9,11)(10,14)(12,13)(15,16)(17,19,23,20)(18,21,25,27)(22,28,30,26)(24,31,32,29);(2,3)(6,7)(11,12)(14,16)(17,20,23,19)(18,21,25,27)(22,28,30,26)(24,29,32,31);(1,3,5,2)(4,7,8,6)(9,12,13,11)(10,14,15,16)(17,21)(18,20)(19,25)(22,29)(23,27)(24,28)(26,32)(30,31);(1,4,5,8)(2,6,3,7)(9,10)(11,16)(12,14)(13,15)(17,22,23,30)(18,24,25,32)(19,26,20,28)(21,29,27,31)Polycyclic presentation:
2^8|a^2=h,b^2=fgh,c^2=fgh,d^2=f,e^2=g,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×#57819, #8101780, #8101965, … |
| maximal normal subgroup of | C2×#57819, #8101780, #8101965, … |
| maximal quotient of | #585432, #586096, #587240, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50807);