Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#389 |
| prime radical | C24 | C22×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
872 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,19,15,20,14,18)(11,21,23,16,25,32,24,33)(12,26,28,36,30,35,29,17)(22,31,37,39,38,40,27,34);(1,3,5,4)(2,6,8,7)(9,11,13,23,15,25,14,24)(10,16,18,21,20,33,19,32)(12,27,28,22,30,37,29,38)(17,34,35,40,36,39,26,31);(1,3,5,4)(2,7,8,6)(9,12,13,28,15,30,14,29)(10,17,18,35,20,36,19,26)(11,22,23,37,25,38,24,27)(16,31,21,34,33,40,32,39);(1,4,5,3)(2,6,8,7)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=e,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,g^b=gi,g^c=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265948);