Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| 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 | |
1829 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,9,11)(7,12,10,8)(15,18)(16,20)(21,22)(23,31)(24,29)(25,30)(26,27)(28,33)(32,36)(34,35);(5,7)(6,8)(9,10)(11,12)(13,14,17,19)(15,16,18,20)(22,27)(24,34)(28,31)(30,32);(1,2,3,4)(7,10)(8,12)(13,15,17,18)(14,16,19,20)(21,23,26,33)(22,28,27,31)(24,32,34,30)(25,35,36,29);(5,6)(7,12)(8,10)(9,11)(13,16)(14,18)(15,19)(17,20)(21,24,26,34)(22,29,27,35)(23,32,33,30)(25,31,36,28);(5,8,9,12)(6,10,11,7)(21,25)(22,30)(23,29)(24,31)(26,36)(27,32)(28,34)(33,35)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=g,d^2=fh,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4293586);