Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30437 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1401 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6,10,15)(7,16,17,26)(8,22,24,13)(9,11,18,14)(12,20,29,21)(19,31,32,36)(23,27,25,30)(28,33,35,34);(1,2)(3,4)(6,11)(7,17)(8,23)(9,18)(12,16)(13,22)(14,15)(19,33)(24,25)(26,29)(28,35)(32,34);(5,7,18,20,10,17,9,21)(6,12,11,26,15,29,14,16)(8,19,23,33,24,32,25,34)(13,28,27,36,22,35,30,31);(2,4)(5,8)(6,13)(7,19)(9,25)(10,24)(11,27)(12,28)(14,30)(15,22)(16,31)(17,32)(18,23)(20,33)(21,34)(26,36)(29,35)Polycyclic presentation:
2^9|a^2=h,c^2=egh,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,g^c=gi,e^d=ei,f^d=fi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 297646);