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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(6,9)(10,12)(13,14,17,19)(15,18,20,16)(21,22,26,24)(23,29,30,27)(25,32,33,28)(31,34,36,35);(13,15,17,20)(14,16,19,18)(22,24)(23,30)(28,32)(31,36);(2,3)(21,23,26,30)(22,27,24,29)(25,31,33,36)(28,34,32,35);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,19)(17,20)(21,24,26,22)(23,27,30,29)(25,32,33,28)(31,34,36,35);(5,7,8,11)(6,10,9,12)(13,16,17,18)(14,15,19,20)(21,25,26,33)(22,28,24,32)(23,31,30,36)(27,34,29,35)Polycyclic presentation:
2^9|a^2=fgh,b^2=fh,c^2=h,d^2=g,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, 4839019);