Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
945 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(22,24)(27,31)(28,32)(34,35);(6,10)(8,16)(11,13)(14,20)(21,22,26,24)(23,27,30,31)(25,32,33,28)(29,35,36,34);(1,2)(3,4)(5,6,9,10)(7,13,15,11)(8,12,16,17)(14,19,20,18)(21,23)(22,27)(24,31)(25,29)(26,30)(28,34)(32,35)(33,36);(5,7,9,15)(6,11,10,13)(8,14,16,20)(12,18,17,19)(21,24,26,22)(23,27,30,31)(25,32,33,28)(29,34,36,35);(5,8)(6,12)(7,14)(9,16)(10,17)(11,18)(13,19)(15,20)(21,25,26,33)(22,28,24,32)(23,29,30,36)(27,34,31,35)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=h,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #482829, #6352850, #6352860, … |
| maximal normal subgroup of | #482829, #6352850, #6352860, … |
| maximal quotient of | #544475, #545052, #545407, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35303);