Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4307937 |
| automorphism | Aut(G) order 4194304 |
| 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 | |
2202 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)(6,9)(10,12)(13,14,18,17)(15,20,19,16)(21,22,27,24)(23,30,25,28)(26,32,34,29)(31,35,33,36);(13,15)(14,16)(17,20)(18,19)(21,23,27,25)(22,28,24,30)(26,31,34,33)(29,35,32,36);(1,2)(3,4)(21,24,27,22)(23,28,25,30)(26,32,34,29)(31,35,33,36);(5,6)(7,10)(8,9)(11,12)(13,16,18,20)(14,19,17,15)(21,25,27,23)(22,28,24,30)(26,33,34,31)(29,35,32,36);(5,7,8,11)(6,10,9,12)(13,17,18,14)(15,20,19,16)(21,26)(22,29)(23,31)(24,32)(25,33)(27,34)(28,35)(30,36)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=h,d^2=f,e^2=i,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, 4215873);