Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19713 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×#127 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
738 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(6,10)(8,16)(11,17)(12,18)(13,20)(15,19)(21,22,25,28)(23,29,30,36)(24,32,33,27)(26,34,35,31);(5,6,8,10,9,12,16,18)(7,13,15,20,14,11,19,17);(5,7)(6,11)(8,15)(9,14)(10,17)(12,13)(16,19)(18,20)(21,23,24,31,25,30,33,34)(22,26,27,29,28,35,32,36);(1,3,4,2)(6,12)(7,14)(10,18)(15,19)(23,30)(26,35)(29,36)(31,34)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=f,d^2=gi,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=dhi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 120424);