Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1852 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)(5,6,9,8)(7,10,12,11)(15,19)(18,20)(21,22)(23,29)(24,28)(25,26)(27,30)(31,35)(32,33)(34,36);(5,7,9,12)(6,10,8,11)(14,17)(18,20)(22,26)(23,30)(28,32)(31,36);(6,8)(10,11)(13,14,16,17)(15,18,19,20)(21,22)(23,29)(24,32)(25,26)(27,30)(28,33)(31,34)(35,36);(1,2)(3,4)(5,8,9,6)(7,11,12,10)(21,23)(22,27)(24,31)(25,30)(26,29)(28,34)(32,35)(33,36);(5,7)(6,10)(8,11)(9,12)(13,15)(14,18)(16,19)(17,20)(21,24,25,33)(22,28,26,32)(23,31,30,36)(27,34,29,35)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=fh,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5121087);