Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1889 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)(5,6,10,9)(7,8,12,11)(15,19)(18,20)(21,22)(23,26)(24,27)(25,28)(29,33)(30,34)(31,32)(35,36);(5,7)(6,11)(8,9)(10,12)(14,17)(18,20)(21,22,25,28)(23,26,30,34)(24,27,32,31)(29,33,36,35);(6,9)(7,12)(13,14,16,17)(15,18,19,20)(21,22)(23,26)(24,31)(25,28)(27,32)(29,35)(30,34)(33,36);(1,2)(3,4)(5,8)(6,7)(9,12)(10,11)(21,23,25,30)(22,26,28,34)(24,29,32,36)(27,33,31,35);(5,9,10,6)(7,8,12,11)(13,15)(14,18)(16,19)(17,20)(21,24,25,32)(22,27,28,31)(23,29,30,36)(26,33,34,35)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=fh,d^2=fg,e^2=f,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, 4514866);