Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2014 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,12)(8,11)(9,10)(14,18)(15,16)(21,22,26,29)(23,27,31,24)(25,33,35,28)(30,32,36,34);(6,10)(11,12)(13,14,17,18)(15,20,16,19)(21,23,26,31)(22,24,29,27)(25,30,35,36)(28,32,33,34);(5,7,9,8)(6,11,10,12)(13,15,17,16)(14,19,18,20)(21,23)(22,27)(24,29)(25,30)(26,31)(28,34)(32,33)(35,36);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,14)(15,19)(16,20)(17,18)(21,24,26,27)(22,23,29,31)(25,34,35,32)(28,36,33,30);(5,8,9,7)(6,11,10,12)(13,16,17,15)(14,19,18,20)(21,25,26,35)(22,28,29,33)(23,30,31,36)(24,32,27,34)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=g,d^2=fg,e^2=f,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, 4769365);