Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
2130 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)(7,9)(8,10)(11,12)(14,18)(15,16)(21,22,26,30)(23,29,25,27)(24,28,34,33)(31,36,32,35);(6,7)(10,11)(13,14)(15,19)(16,20)(17,18)(21,23,26,25)(22,27,30,29)(24,31,34,32)(28,35,33,36);(5,6,9,7)(8,11,12,10)(13,15,17,16)(14,19,18,20)(21,24)(22,28)(23,31)(25,32)(26,34)(27,35)(29,36)(30,33);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,16,17,15)(14,19,18,20)(21,25,26,23)(22,29,30,27)(24,32,34,31)(28,36,33,35);(5,8)(6,10)(7,11)(9,12)(13,14)(15,20)(16,19)(17,18)(21,22)(23,27)(24,33)(25,29)(26,30)(28,34)(31,36)(32,35)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=g,d^2=i,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, 4571739);