Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1361 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(17,18,22,23)(19,28,29,41)(20,32,35,25)(21,26,37,27)(24,39,38,31)(30,46,33,45)(34,44,36,43)(40,48,42,47);(1,2,4,5)(3,6,7,8)(9,10,13,16)(11,15,12,14)(18,23)(19,29)(25,32)(26,27)(30,33)(31,39)(43,44)(47,48);(2,5)(6,8)(9,11,13,12)(10,14,16,15)(17,19,21,31,22,29,37,39)(18,24,27,28,23,38,26,41)(20,33,36,47,35,30,34,48)(25,42,43,46,32,40,44,45);(1,3,4,7)(2,6,5,8)(17,20,22,35)(18,25,23,32)(19,30,29,33)(21,36,37,34)(24,40,38,42)(26,44,27,43)(28,45,41,46)(31,48,39,47);(9,12,13,11)(10,15,16,14)(17,21,22,37)(18,26,23,27)(19,31,29,39)(20,34,35,36)(24,41,38,28)(25,43,32,44)(30,47,33,48)(40,45,42,46)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=fgh,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6761784);