Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
2471 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(6,7)(8,12)(15,19)(18,20)(22,27)(24,31)(25,33)(30,35);(6,7)(10,11)(14,17)(18,20)(21,22,26,27)(23,25,29,33)(24,28,31,34)(30,32,35,36);(5,6,9,7)(8,10,12,11)(13,14,16,17)(15,18,19,20)(21,23)(22,25)(24,30)(26,29)(27,33)(28,32)(31,35)(34,36);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(21,24)(22,28)(23,30)(25,32)(26,31)(27,34)(29,35)(33,36);(5,8)(6,10)(7,11)(9,12)(13,15)(14,18)(16,19)(17,20)(21,25)(22,29)(23,27)(24,32)(26,33)(28,35)(30,34)(31,36)Polycyclic presentation:
2^9|a^2=fgi,b^2=f,c^2=h,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4514818);