Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420420 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60454 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1690 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,12,8,11,16,10,15)(7,17,19,13,21,22,20,30)(9,24,26,32,28,23,27,14)(18,29,34,31,33,36,25,35);(1,2)(3,4)(5,7)(6,13)(8,22)(9,25)(10,20)(11,21)(12,19)(14,31)(15,17)(16,30)(18,26)(23,29)(24,36)(27,33)(28,34)(32,35);(2,4)(5,8,10,16,11,15,12,6)(7,13,19,17,21,30,20,22)(9,23,26,32,28,24,27,14)(18,29,25,31,33,36,34,35);(5,9,10,26,11,28,12,27)(6,14,8,23,16,32,15,24)(7,18,19,25,21,33,20,34)(13,29,17,31,30,36,22,35)Polycyclic presentation:
2^9|a^2=ef,c^2=e,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,f^c=fi,h^c=hi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327616);