Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2330 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)(5,6,9,8)(7,11,12,10)(13,14)(15,19)(16,17)(18,20)(22,23)(27,29)(28,30)(34,35);(13,15)(14,18)(16,19)(17,20)(21,22,26,23)(24,27,32,29)(25,30,33,28)(31,35,36,34);(1,2)(3,4)(15,20)(18,19)(21,23,26,22)(24,27,32,29)(25,30,33,28)(31,34,36,35);(5,7,9,12)(6,10,8,11)(13,15)(14,19)(16,18)(17,20)(21,24,26,32)(22,27,23,29)(25,31,33,36)(28,34,30,35);(5,8,9,6)(7,11,12,10)(13,16,17,14)(15,18,20,19)(21,25)(22,28)(23,30)(24,31)(26,33)(27,34)(29,35)(32,36)Polycyclic presentation:
2^9|a^2=i,b^2=fh,c^2=fh,d^2=fhi,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4286892);