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) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
959 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,10,13)(7,8,14,11)(9,12,17,19)(15,16,20,18)(22,26)(23,29)(24,31)(34,36);(7,14)(8,11)(9,17)(12,19)(21,22,25,26)(23,27,29,33)(24,32,31,28)(30,36,35,34);(1,2)(3,4)(5,7)(6,11)(8,13)(9,15)(10,14)(12,18)(16,19)(17,20)(21,22)(23,27)(24,28)(25,26)(29,33)(30,34)(31,32)(35,36);(5,8,10,11)(6,7,13,14)(9,16,17,18)(12,15,19,20)(21,23)(22,27)(24,30)(25,29)(26,33)(28,34)(31,35)(32,36);(5,9)(6,12)(7,15)(8,16)(10,17)(11,18)(13,19)(14,20)(21,24,25,31)(22,28,26,32)(23,30,29,35)(27,34,33,36)Polycyclic presentation:
2^8|a^2=fh,b^2=f,d^2=fh,e^2=f,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 | #98464, #246777, #246778, … |
| maximal normal subgroup of | #98464, #246777, #246778, … |
| maximal quotient of | #557045, #557052, #557367, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39123);