Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437764 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1502 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,8)(4,6)(5,7)(11,15)(13,16)(17,18,22,26,23,27,21,34)(19,28,30,39,31,38,36,24)(20,25,32,42,33,41,35,40)(29,43,46,47,45,48,44,37);(9,10)(11,16)(12,14)(13,15)(17,19,23,31)(18,24,27,39)(20,29,33,45)(21,30,22,36)(25,37,41,47)(26,38,34,28)(32,44,35,46)(40,43,42,48);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,20,23,33)(18,25,27,41)(19,29,31,45)(21,35,22,32)(24,37,39,47)(26,42,34,40)(28,43,38,48)(30,46,36,44);(10,14)(13,16)(17,21,23,22)(18,26,27,34)(19,30,31,36)(20,32,33,35)(24,38,39,28)(25,40,41,42)(29,44,45,46)(37,43,47,48);(1,4,5,3)(2,6,7,8)(20,33)(25,41)(29,45)(32,35)(37,47)(40,42)(43,48)(44,46)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gi,d^2=i,e^2=gh,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6376376);