Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4307937 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2178 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)(5,6,8,10)(7,9,11,12)(13,14,19,18)(15,20,17,16)(22,26)(23,29)(24,31)(34,36);(5,6)(7,9)(8,10)(11,12)(13,15,19,17)(14,16,18,20)(22,26)(27,32)(28,33)(34,36);(13,16)(14,17)(15,18)(19,20)(21,22,25,26)(23,27,29,32)(24,28,31,33)(30,34,35,36);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,17,19,15)(14,20,18,16)(21,23,25,29)(22,27,26,32)(24,30,31,35)(28,34,33,36);(5,7,8,11)(6,9,10,12)(13,18,19,14)(15,20,17,16)(21,24)(22,28)(23,30)(25,31)(26,33)(27,34)(29,35)(32,36)Polycyclic presentation:
2^9|a^2=f,b^2=gh,c^2=h,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4215235);