Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| 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 | |
2186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(6,9)(10,12)(13,14)(15,18)(16,19)(17,20)(21,22)(23,29)(24,32)(25,26)(27,30)(28,33)(31,34)(35,36);(13,14,17,20)(15,18,16,19)(22,26)(23,30)(28,32)(31,36);(2,3)(13,15,17,16)(14,18,20,19)(21,22)(23,27)(24,28)(25,26)(29,30)(31,34)(32,33)(35,36);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,19)(17,20)(21,23,25,30)(22,27,26,29)(24,31,33,36)(28,34,32,35);(5,7,8,11)(6,10,9,12)(13,16,17,15)(14,19,20,18)(21,24)(22,28)(23,31)(25,33)(26,32)(27,34)(29,35)(30,36)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=fh,d^2=gh,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4839350);