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 | C2×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
865 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,15)(14,16)(17,18)(19,27)(20,25)(21,31)(22,23)(24,28)(26,32)(29,30);(2,4)(6,7)(9,11,14,12)(10,15,16,13)(18,23)(19,28)(20,30)(21,32);(1,3,5,8)(2,6,4,7)(9,12,14,11)(10,15,16,13)(17,19)(18,24)(20,31)(21,29)(22,28)(23,27)(25,32)(26,30);(1,4,5,2)(3,7,8,6)(9,13,14,15)(10,11,16,12)(17,20,22,30)(18,25,23,29)(19,26,28,31)(21,27,32,24);(17,21)(18,26)(19,29)(20,27)(22,32)(23,31)(24,30)(25,28)Polycyclic presentation:
2^8|a^2=gh,b^2=fg,c^2=fh,d^2=f,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 | #130902, #254389, #254390, … |
| maximal normal subgroup of | #130902, #254389, #254390, … |
| maximal quotient of | #535315, #535536, #535941, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 37453);