Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| automorphism | Aut(G) order 2097152 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2220 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)(7,11)(10,12)(13,14,17,20)(15,18,16,19)(21,22)(23,29)(24,31)(25,28)(26,27)(30,36)(32,33)(34,35);(6,9)(10,12)(13,15,17,16)(14,18,20,19)(22,27)(24,29)(28,35)(32,36);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,19)(17,20)(21,23)(22,24)(25,33)(26,31)(27,29)(28,36)(30,34)(32,35);(1,2)(3,4)(13,16,17,15)(14,19,20,18)(21,24)(22,23)(25,32)(26,29)(27,31)(28,30)(33,35)(34,36);(5,7)(6,10)(8,11)(9,12)(21,25)(22,28)(23,30)(24,32)(26,34)(27,35)(29,36)(31,33)Polycyclic presentation:
2^9|a^2=fg,b^2=gi,d^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4298483);