Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#400 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,16,11,14,9,15)(7,12,19,27,18,26,23,31)(8,13,22,30,21,29,24,32)(17,25,34,36,33,35,20,28);(1,2)(3,4)(5,7)(6,12,14,26)(8,20,21,34)(9,19)(10,23)(11,18)(13,28)(15,27,16,31)(17,22,33,24)(25,32)(29,36)(30,35);(2,4)(5,8)(6,13,14,29)(7,17,18,33)(9,22)(10,24)(11,21)(12,25)(15,30,16,32)(19,20,23,34)(26,35)(27,28)(31,36);(7,18)(12,26)(17,33)(19,23)(20,34)(25,35)(27,31)(28,36)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,d^b=di,e^b=ei,h^b=hi,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265446);