Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(6,9)(10,12)(15,19)(18,20)(21,22,27,30)(23,24,25,28)(26,34,35,29)(31,36,33,32);(14,17)(18,20)(21,23,27,25)(22,28,30,24)(26,33,35,31)(29,32,34,36);(5,6)(7,10)(8,9)(11,12)(13,14,16,17)(15,18,19,20)(21,24,27,28)(22,23,30,25)(26,32,35,36)(29,31,34,33);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(21,25,27,23)(22,24,30,28)(26,33,35,31)(29,32,34,36);(5,7,8,11)(6,10,9,12)(13,15)(14,18)(16,19)(17,20)(21,26,27,35)(22,29,30,34)(23,31,25,33)(24,32,28,36)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=h,d^2=fgi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4249105);