Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5020720 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1988 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(15,19)(17,20)(21,22,27,30)(23,31,24,28)(25,26,36,29)(32,35,34,33);(5,7)(6,9)(8,12)(10,11)(21,23,27,24)(22,28,30,31)(25,32,36,34)(26,33,29,35);(2,3)(6,8)(7,11)(13,14,16,18)(15,17,19,20)(21,24,27,23)(22,28,30,31)(25,34,36,32)(26,33,29,35);(1,3,4,2)(5,8,10,6)(7,12,11,9)(13,15)(14,17)(16,19)(18,20)(21,25)(22,29)(23,32)(24,34)(26,30)(27,36)(28,35)(31,33);(5,9,10,12)(6,11,8,7)(21,26,27,29)(22,25,30,36)(23,33,24,35)(28,32,31,34)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=gi,d^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^c=dg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4931820);