Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1966 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6)(7,9)(8,10)(11,12)(15,19)(18,20)(21,22,26,27)(23,31,32,28)(24,29,34,35)(25,33,36,30);(1,2,3,4)(6,7)(10,11)(14,17)(18,20)(22,27)(23,32)(29,35)(30,33);(1,2)(3,4)(8,12)(10,11)(13,14)(15,18)(16,17)(19,20)(21,23,26,32)(22,28,27,31)(24,33,34,30)(25,35,36,29);(5,7,9,6)(8,10,12,11)(13,15)(14,18)(16,19)(17,20)(21,24)(22,29)(23,33)(25,28)(26,34)(27,35)(30,32)(31,36);(5,8,9,12)(6,10,7,11)(21,25)(22,30)(23,29)(24,31)(26,36)(27,33)(28,34)(32,35)Polycyclic presentation:
2^9|a^2=g,b^2=fhi,c^2=g,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3275683);