Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #201384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#392 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1419 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,25,28,20,16)(3,14,18,32,27,22,29,5)(4,10,21,11,15,30,23,17)(7,13,26,31,8,9,24,19);(1,3,15,31,8,17,22,16)(2,6,18,23,9,24,10,5)(4,19,7,11,14,12,25,27)(13,26,30,32,28,20,29,21);(2,9)(5,23)(6,24)(10,18)(13,28)(20,26)(21,32)(29,30);(1,4,8,14)(2,10,9,18)(3,16,17,31)(5,20,23,26)(6,21,24,32)(7,22,25,15)(11,19,27,12)(13,29,28,30)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=i,e^2=h,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,f^b=fi,e^c=ei,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72213685, #72213768, #72213800, … |
| maximal normal subgroup of | #72213685, #72213768, #72213800, … |
| maximal quotient of | C2×#201616, #72240257, #72240264, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 108812);