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 | |
1897 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)(17,20)(21,22,26,27)(23,28,30,25)(24,32,33,29)(31,35,34,36);(5,7)(6,10)(8,11)(9,12)(13,14,16,18)(15,17,19,20)(22,27)(23,30)(24,33)(35,36);(5,6,9,11)(7,10,12,8)(13,14)(15,17)(16,18)(19,20)(21,23)(22,28)(24,34)(25,27)(26,30)(29,35)(31,33)(32,36);(1,2,3,4)(5,8)(6,7)(9,10)(11,12)(21,24,26,33)(22,29,27,32)(23,31,30,34)(25,35,28,36);(5,6)(7,8)(9,11)(10,12)(13,15,16,19)(14,17,18,20)(21,25,26,28)(22,30,27,23)(24,35,33,36)(29,34,32,31)Polycyclic presentation:
2^9|a^2=fg,b^2=fgh,c^2=g,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, 4990486);