Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
2638 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,14,16,18)(15,17,19,20)(22,27)(24,29)(28,33)(32,35);(5,6)(7,9)(8,10)(11,12)(15,19)(17,20)(21,22,26,27)(23,29,31,24)(25,33,34,28)(30,32,36,35);(5,6,8,10)(7,9,12,11)(13,15,16,19)(14,17,18,20)(21,23,26,31)(22,24,27,29)(25,30,34,36)(28,32,33,35);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(21,24)(22,23)(25,32)(26,29)(27,31)(28,30)(33,36)(34,35);(5,7)(6,9)(8,12)(10,11)(13,14)(15,17)(16,18)(19,20)(21,25,26,34)(22,28,27,33)(23,30,31,36)(24,32,29,35)Polycyclic presentation:
2^9|a^2=ghi,b^2=f,c^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4849150);