Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1668 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)(5,6,8,10)(7,11,12,9)(15,19)(18,20)(21,22,27,32)(23,29,24,28)(25,34,35,30)(26,33,36,31);(5,6)(7,9)(8,10)(11,12)(14,17)(18,20)(21,23,27,24)(22,28,32,29)(25,33,35,31)(26,30,36,34);(7,12)(9,11)(13,14)(15,18)(16,17)(19,20)(21,24,27,23)(22,29,32,28)(25,33,35,31)(26,30,36,34);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(21,25,27,35)(22,30,32,34)(23,31,24,33)(26,29,36,28);(5,7,8,12)(6,9,10,11)(13,15)(14,18)(16,19)(17,20)(21,26,27,36)(22,31,32,33)(23,30,24,34)(25,29,35,28)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=g,d^2=fgi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4571877);