Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | Aut(G) order 65536 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
796 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(6,8)(10,11)(14,15)(17,18,22,20)(19,25,26,23)(21,28,29,24)(27,30,32,31);(9,10,13,11)(12,15,16,14)(19,26)(21,29)(23,25)(24,28);(1,2,4,5)(3,6,7,8)(9,11,13,10)(12,15,16,14)(17,19,22,26)(18,23,20,25)(21,27,29,32)(24,30,28,31);(1,3,4,7)(2,6,5,8)(9,12)(10,14)(11,15)(13,16)(17,20,22,18)(19,23,26,25)(21,28,29,24)(27,30,32,31);(17,21,22,29)(18,24,20,28)(19,27,26,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=g,d^2=fg,e^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #7093292, #7093294, #7093300, … |
| maximal normal subgroup of | #7093292, #7093294, #7093300, … |
| maximal quotient of | #535457, #535559, #536064, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38189);