Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23444 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#385 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1634 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,14,10,13,22,28)(7,11,17,25,16,24,20,29)(8,12,19,27,18,26,21,30)(15,23,32,35,31,34,33,36);(1,2)(3,4)(5,7)(6,11,13,24)(8,15,18,31)(9,20)(10,16)(12,23)(14,29,28,25)(17,22)(19,33,21,32)(26,34)(27,36)(30,35);(2,4)(5,8)(6,12,13,26)(7,15,16,31)(9,21)(10,18)(11,23)(14,30,28,27)(17,33,20,32)(19,22)(24,34)(25,36)(29,35);(7,16)(11,24)(15,31)(17,20)(23,34)(25,29)(32,33)(35,36)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,d^b=di,e^b=ei,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265452);