Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| 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 | |
2249 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)(5,6,8,10)(7,9,11,12)(15,19)(17,20)(22,26)(27,33)(28,31)(34,35);(13,14,16,18)(15,17,19,20)(21,22)(23,27)(24,28)(25,26)(29,34)(30,33)(31,32)(35,36);(5,6)(7,9)(8,10)(11,12)(13,14)(15,17)(16,18)(19,20)(21,23,25,30)(22,27,26,33)(24,29,32,36)(28,34,31,35);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(21,22)(23,27)(24,31)(25,26)(28,32)(29,35)(30,33)(34,36);(5,7,8,11)(6,9,10,12)(13,15)(14,17)(16,19)(18,20)(21,24)(22,28)(23,29)(25,32)(26,31)(27,34)(30,36)(33,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4325847);