Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| 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 | |
2062 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)(7,12)(8,11)(15,19)(18,20)(21,22,26,27)(23,28,30,24)(25,33,35,29)(31,32,36,34);(5,6,10,9)(7,11,12,8)(14,17)(18,20)(22,27)(23,30)(29,33)(31,36);(5,7,10,12)(6,11,9,8)(13,14)(15,18)(16,17)(19,20)(21,23)(22,28)(24,27)(25,31)(26,30)(29,34)(32,33)(35,36);(1,2)(3,4)(5,8)(6,7)(9,12)(10,11)(21,24,26,28)(22,23,27,30)(25,34,35,32)(29,36,33,31);(5,9,10,6)(7,8,12,11)(13,15)(14,18)(16,19)(17,20)(21,25,26,35)(22,29,27,33)(23,31,30,36)(24,32,28,34)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=g,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4325633);