Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2026 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(6,9)(10,12)(14,15)(16,20)(21,22,26,29)(23,24,31,27)(25,28,34,36)(30,32,33,35);(2,4)(14,15)(18,19)(21,23,26,31)(22,27,29,24)(25,33,34,30)(28,32,36,35);(1,2,3,4)(13,14,17,15)(16,18,20,19)(21,24)(22,23)(25,32)(26,27)(28,30)(29,31)(33,36)(34,35);(5,6,8,9)(7,10,11,12)(13,15,17,14)(16,19,20,18)(21,23)(22,27)(24,29)(25,33)(26,31)(28,32)(30,34)(35,36);(5,7,8,11)(6,10,9,12)(13,16)(14,18)(15,19)(17,20)(21,25,26,34)(22,28,29,36)(23,30,31,33)(24,32,27,35)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=h,d^2=fi,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3179012);