Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806671 |
| 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 | |
1349 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)(19,24)(20,25)(21,35)(22,27)(23,28)(26,36)(29,37)(30,38)(31,39)(32,40)(33,41)(34,43)(42,46)(44,47)(45,48);(9,10)(11,14)(12,16)(13,15)(17,19,23,30)(18,24,28,38)(20,31,33,46)(21,22,36,32)(25,39,41,42)(26,40,35,27)(29,44,45,34)(37,47,48,43);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,20,32,44,23,33,22,34)(18,25,40,47,28,41,27,43)(19,29,36,46,30,45,21,31)(24,37,26,42,38,48,35,39);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,14,15,16)(20,33)(25,41)(29,45)(31,46)(34,44)(37,48)(39,42)(43,47);(1,4,5,3)(2,8,6,7)(9,12,13,11)(10,14,15,16)(17,21,22,30,23,36,32,19)(18,26,27,24,28,35,40,38)(20,31,34,45,33,46,44,29)(25,42,43,37,41,39,47,48)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fhi,d^2=g,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6747645);