Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| 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 | |
730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,7)(6,8)(11,15)(14,16)(17,18,22,19)(20,25,28,23)(21,26,29,24)(27,30,32,31);(2,5)(3,7)(9,10,13,12)(11,14,15,16)(18,19)(21,29)(23,25)(27,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,16)(13,15)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30);(1,3,4,7)(2,6,5,8)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32);(1,2)(3,6)(4,5)(7,8)(9,12,13,10)(11,16,15,14)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=f,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #7223598, #7223600, #7223622, … |
| maximal normal subgroup of | #7223598, #7223600, #7223622, … |
| maximal quotient of | #557334, #557335, #557539, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39070);