Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1364 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)(3,7)(4,8)(5,6)(9,10,13,15)(11,16,12,14)(17,18,23,28)(19,24,31,39)(20,25,34,42)(21,26,30,38)(22,27,33,41)(29,37,46,48)(32,40,35,43)(36,44,45,47);(9,10)(11,14)(12,16)(13,15)(17,19,23,31)(18,24,28,39)(20,32,34,35)(21,33,30,22)(25,40,42,43)(26,41,38,27)(29,45,46,36)(37,47,48,44);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,20,33,45,23,34,22,36)(18,25,41,47,28,42,27,44)(19,29,21,35,31,46,30,32)(24,37,26,43,39,48,38,40);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,14,15,16)(20,34)(25,42)(29,46)(32,35)(36,45)(37,48)(40,43)(44,47);(1,4,5,3)(2,8,6,7)(9,12,13,11)(10,14,15,16)(17,21,22,19,23,30,33,31)(18,26,27,24,28,38,41,39)(20,35,36,29,34,32,45,46)(25,43,44,37,42,40,47,48)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fhi,d^2=g,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6747533);