Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | |
2213 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)(7,11)(8,12)(9,10)(13,14)(15,19)(16,17)(18,20)(21,22,26,24)(23,27,29,30)(25,28,34,32)(31,35,33,36);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,14,17,16)(15,18,20,19)(22,24)(23,29)(28,32)(31,33);(1,2,3,4)(6,10)(8,11)(14,16)(18,19)(21,23)(22,27)(24,30)(25,33)(26,29)(28,36)(31,34)(32,35);(5,7)(6,8)(9,12)(10,11)(13,15)(14,18)(16,19)(17,20)(21,24,26,22)(23,30,29,27)(25,32,34,28)(31,36,33,35);(1,2)(3,4)(5,8)(6,7)(9,11)(10,12)(13,16,17,14)(15,19,20,18)(21,25,26,34)(22,28,24,32)(23,31,29,33)(27,35,30,36)Polycyclic presentation:
2^9|a^2=fg,b^2=fhi,c^2=gi,d^2=fg,e^2=ghi,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, 2459701);