Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #917169 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
1951 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)(6,7)(10,11)(21,22,27,30)(23,31,24,28)(25,36,29,26)(32,33,34,35);(6,7)(10,11)(14,17)(18,20)(21,23,27,24)(22,28,30,31)(25,32,29,34)(26,33,36,35);(5,6,9,7)(8,10,12,11)(13,14)(15,18)(16,17)(19,20)(21,24,27,23)(22,28,30,31)(25,32,29,34)(26,35,36,33);(1,2,3,4)(5,7,9,6)(8,11,12,10)(21,25,27,29)(22,26,30,36)(23,32,24,34)(28,33,31,35);(5,8,9,12)(6,10,7,11)(13,15,16,19)(14,18,17,20)(21,26)(22,29)(23,33)(24,35)(25,30)(27,36)(28,34)(31,32)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=fg,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 855192);