Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | |
1892 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)(7,11)(10,12)(16,20)(18,19)(21,22,26,28)(23,29,30,24)(25,33,34,27)(31,32,36,35);(1,2)(3,4)(6,9)(10,12)(13,14,17,15)(16,18,20,19)(23,30)(24,29)(31,36)(32,35);(1,2,3,4)(5,6,8,9)(7,10,11,12)(13,15,17,14)(16,19,20,18)(21,23)(22,24)(25,31)(26,30)(27,32)(28,29)(33,35)(34,36);(5,7,8,11)(6,10,9,12)(13,16)(14,18)(15,19)(17,20)(21,24)(22,23)(25,32)(26,29)(27,31)(28,30)(33,36)(34,35);(1,2)(3,4)(21,25,26,34)(22,27,28,33)(23,31,30,36)(24,32,29,35)Polycyclic presentation:
2^9|a^2=fg,b^2=ghi,c^2=fi,d^2=gh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2167966);