Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| 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 | |
2064 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)(7,12)(8,10)(15,19)(18,20)(21,22,25,29)(23,27,31,26)(24,28,33,36)(30,35,32,34);(5,6,9,11)(7,10,12,8)(14,17)(18,20)(21,23,25,31)(22,26,29,27)(24,32,33,30)(28,35,36,34);(5,7,9,12)(6,8,11,10)(13,14)(15,18)(16,17)(19,20)(21,23)(22,27)(24,30)(25,31)(26,29)(28,35)(32,33)(34,36);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(21,22)(23,26)(24,28)(25,29)(27,31)(30,34)(32,35)(33,36);(5,8,9,10)(6,7,11,12)(13,15,16,19)(14,18,17,20)(21,24)(22,28)(23,30)(25,33)(26,34)(27,35)(29,36)(31,32)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=g,d^2=fgi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4894679);