Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24802 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#389 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1306 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)(5,6,10,14,11,16,9,15)(7,12,19,27,20,29,18,28)(8,21,24,31,25,32,23,13)(17,30,22,35,34,36,33,26);(2,4)(5,7,9,18,11,20,10,19)(6,12,15,28,16,29,14,27)(8,22,24,34,25,33,23,17)(13,30,21,35,31,36,32,26);(5,8,9,23,11,25,10,24)(6,13,14,21,16,31,15,32)(7,17,18,33,20,34,19,22)(12,26,27,30,29,35,28,36);(2,4)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72250519, #72250529, #72250537, … |
| maximal normal subgroup of | #72250519, #72250529, #72250537, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269276);