Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1290 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)(3,6)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,27)(20,25)(21,23)(22,26)(24,31)(28,38)(29,37)(30,42)(32,45)(33,39)(34,36)(35,41)(40,44)(43,47)(46,48);(1,3,6,2)(4,8,5,7)(18,23)(19,28)(20,32)(22,29)(24,38)(25,41)(27,40)(31,44)(34,35)(36,45)(39,42)(43,46);(10,13)(14,16)(17,19,29,44,21,31,22,28)(18,24,37,38,23,27,26,40)(20,33,32,43,34,30,35,46)(25,42,45,48,36,39,41,47);(1,4,6,5)(2,7,3,8)(9,11,12,15)(10,14,13,16)(18,23)(20,34)(24,27)(26,37)(30,33)(32,35)(38,40)(43,46);(1,5,6,4)(2,7,3,8)(17,20,22,35,21,34,29,32)(18,25,26,41,23,36,37,45)(19,30,28,43,31,33,44,46)(24,39,40,48,27,42,38,47)Polycyclic presentation:
2^9|b^2=fi,c^2=hi,d^2=fg,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6735368);