Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2289 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6)(7,11)(8,10)(9,12)(15,19)(18,20)(21,22)(23,26)(24,29)(25,28)(27,31)(30,33)(32,36)(34,35);(2,4)(5,7,10,9)(6,11,8,12)(22,23)(24,31)(25,33)(35,36);(1,2)(3,4)(5,6,10,8)(7,12,9,11)(13,14,17,16)(15,18,19,20)(21,23,26,22)(24,27,31,29)(25,30,33,28)(32,35,34,36);(5,8,10,6)(7,11,9,12)(13,15,17,19)(14,18,16,20)(21,24)(22,27)(23,29)(25,34)(26,31)(28,36)(30,35)(32,33);(5,9,10,7)(6,11,8,12)(13,16,17,14)(15,20,19,18)(21,25)(22,28)(23,30)(24,32)(26,33)(27,35)(29,36)(31,34)Polycyclic presentation:
2^9|b^2=fg,c^2=i,d^2=fi,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3207235);