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 | |
1893 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,11,10)(7,8,12,9)(15,19)(18,20)(21,22)(23,27)(24,26)(25,32)(28,31)(29,33)(30,35)(34,36);(5,7)(6,9)(8,10)(11,12)(14,17)(18,20)(21,23,26,31)(22,27,24,28)(25,30,33,36)(29,34,32,35);(5,8,11,9)(6,12,10,7)(13,14)(15,18)(16,17)(19,20)(21,23)(22,28)(24,27)(25,30)(26,31)(29,35)(32,34)(33,36);(1,2)(3,4)(5,9,11,8)(6,12,10,7)(21,24,26,22)(23,27,31,28)(25,32,33,29)(30,34,36,35);(5,10,11,6)(7,8,12,9)(13,15)(14,18)(16,19)(17,20)(21,25)(22,29)(23,30)(24,32)(26,33)(27,34)(28,35)(31,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,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, 4505257);