Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2069 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #396 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#396 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1726 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)(5,6,10,15,11,16,9,14)(7,17,20,29,21,28,19,12)(8,22,24,32,26,31,25,13)(18,33,23,30,35,27,34,36);(5,7,9,19,11,21,10,20)(6,12,15,17,16,29,14,28)(8,23,24,35,26,34,25,18)(13,30,31,33,32,36,22,27);(2,4)(5,8,9,24,11,26,10,25)(6,13,15,31,16,32,14,22)(7,18,19,23,21,35,20,34)(12,27,17,30,29,33,28,36);(5,9,11,10)(6,14,16,15)(7,19,21,20)(8,24,26,25)(12,28,29,17)(13,22,32,31)(18,23,35,34)(27,36,33,30)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=e,d^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,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, 265305);