Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| 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 | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
855 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(6,8)(10,13)(14,16)(17,18,22,20)(19,25,26,23)(21,24,29,28)(27,31,32,30);(9,10,12,13)(11,14,15,16)(19,26)(23,25)(27,32)(30,31);(1,2)(3,6)(4,5)(7,8)(17,19)(18,23)(20,25)(21,27)(22,26)(24,30)(28,31)(29,32);(1,3,4,7)(2,6,5,8)(9,11,12,15)(10,14,13,16)(17,20,22,18)(19,25,26,23)(21,24,29,28)(27,30,32,31);(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)Polycyclic presentation:
2^8|a^2=h,b^2=f,d^2=fg,b^a=bf,c^a=cg,c^b=ch,e^d=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #129838, #253261, #253263, … |
| maximal normal subgroup of | #129838, #253261, #253263, … |
| maximal quotient of | #517633, #517638, #517674, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29001);