Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
771 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)(3,7)(4,5)(6,8)(9,10,14,15)(11,13,12,16);(1,3,5,8)(2,6,4,7)(10,15)(11,12)(18,22)(20,28)(23,25)(26,32);(3,8)(6,7)(9,11,14,12)(10,16,15,13)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32);(1,3)(2,7)(4,6)(5,8)(9,12,14,11)(10,13,15,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31);(1,4,5,2)(3,6,8,7)(9,13)(10,11)(12,15)(14,16)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=g,b^2=fg,c^2=g,d^2=fgh,e^2=gh,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 | #132779, #487524, #488263, … |
| maximal normal subgroup of | #132779, #487524, #488263, … |
| maximal quotient of | #585377, #585707, #587650, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51976);