Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4308740 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2190 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)(7,11)(9,12)(14,17)(15,19)(21,22,25,28)(23,29,31,26)(24,27,32,34)(30,35,36,33);(5,6,8,10)(7,9,11,12)(21,22)(23,26)(24,27)(25,28)(29,31)(30,33)(32,34)(35,36);(5,6)(7,9)(8,10)(11,12)(13,14)(15,18)(16,17)(19,20)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36);(1,2)(3,4)(13,15,16,19)(14,18,17,20)(21,23,25,31)(22,26,28,29)(24,30,32,36)(27,33,34,35);(5,7)(6,9)(8,11)(10,12)(21,24,25,32)(22,27,28,34)(23,30,31,36)(26,33,29,35)Polycyclic presentation:
2^9|a^2=f,b^2=h,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4216136);