Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667604 |
| 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 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1518 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)(4,6)(5,8)(9,10,12,14)(11,15,16,13)(21,34)(26,40)(31,45)(33,36)(35,41)(38,47)(43,48)(44,46);(1,3)(2,5)(4,8)(6,7)(17,18,23,28)(19,29,32,42)(20,27,22,25)(21,35,34,41)(24,37,39,30)(26,36,40,33)(31,38,45,47)(43,44,48,46);(9,11,12,16)(10,13,14,15)(17,19,23,32)(18,24,28,39)(20,30,22,37)(21,31,34,45)(25,29,27,42)(26,38,40,47)(33,44,36,46)(35,43,41,48);(2,6)(5,7)(17,20,23,22)(18,25,28,27)(19,30,32,37)(21,36,34,33)(24,29,39,42)(26,41,40,35)(31,46,45,44)(38,48,47,43);(3,8)(5,7)(17,21,23,34)(18,26,28,40)(19,31,32,45)(20,33,22,36)(24,38,39,47)(25,35,27,41)(29,43,42,48)(30,44,37,46)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gi,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6595231);