Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1899 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)(7,8)(9,11)(10,12)(15,19)(18,20)(21,22,26,27)(23,29,31,25)(24,32,33,28)(30,34,36,35);(5,7,9,12)(6,10,11,8)(14,17)(18,20)(22,27)(24,33)(25,29)(30,36);(5,7,9,12)(6,8,11,10)(13,14)(15,18)(16,17)(19,20)(21,23,26,31)(22,25,27,29)(24,30,33,36)(28,34,32,35);(1,2,3,4)(5,7)(6,8)(9,12)(10,11)(21,24,26,33)(22,28,27,32)(23,30,31,36)(25,34,29,35);(5,8)(6,7)(9,10)(11,12)(13,15,16,19)(14,18,17,20)(21,25)(22,23)(24,34)(26,29)(27,31)(28,30)(32,36)(33,35)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=f,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4989967);