Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394590 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2622 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1149 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,12,29,11,17,10,16)(7,18,24,13,21,35,20,31)(8,22,19,32,26,14,25,36)(9,15,28,30,23,34,27,33);(1,2)(3,4)(5,7,10,24,11,21,12,20)(6,13,16,35,17,31,29,18)(8,23,25,27,26,9,19,28)(14,30,32,34,22,33,36,15);(2,4)(5,8,12,25,11,26,10,19)(6,14,16,36,17,22,29,32)(7,9,24,27,21,23,20,28)(13,30,18,34,31,33,35,15);(5,9)(6,15)(7,19)(8,24)(10,28)(11,23)(12,27)(13,14)(16,30)(17,34)(18,36)(20,26)(21,25)(22,31)(29,33)(32,35)Polycyclic presentation:
2^9|a^2=ei,b^2=efh,c^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 314277);