Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2083 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)(6,9)(10,11)(15,19)(18,20)(21,22,26,28)(23,29,31,24)(25,33,35,27)(30,32,36,34);(5,6)(7,11)(8,9)(10,12)(14,17)(18,20)(21,23,26,31)(22,24,28,29)(25,30,35,36)(27,32,33,34);(5,7)(6,10)(8,12)(9,11)(13,14,16,17)(15,18,19,20)(21,23)(22,24)(25,30)(26,31)(27,32)(28,29)(33,34)(35,36);(1,2)(3,4)(5,7)(6,11)(8,12)(9,10)(21,24)(22,23)(25,34)(26,29)(27,36)(28,31)(30,33)(32,35);(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,19)(17,20)(21,25,26,35)(22,27,28,33)(23,30,31,36)(24,32,29,34)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4770048);