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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2214 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)(5,6)(7,11)(8,10)(9,12)(13,14,17,16)(15,19,20,18)(21,22)(23,26)(24,29)(25,30)(27,32)(28,33)(31,34)(35,36);(5,6,8,10)(7,9,12,11)(14,16)(18,19)(22,23)(25,33)(27,29)(31,36);(7,12)(9,11)(13,15,17,20)(14,18,16,19)(21,23,26,22)(24,27,32,29)(25,30,33,28)(31,34,36,35);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15)(14,18)(16,19)(17,20)(21,24,26,32)(22,27,23,29)(25,31,33,36)(28,34,30,35);(5,6)(7,9)(8,10)(11,12)(13,16,17,14)(15,19,20,18)(21,25)(22,28)(23,30)(24,31)(26,33)(27,34)(29,35)(32,36)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=g,d^2=fgh,e^2=gh,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, 4227160);