Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535823 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1284 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,6)(5,8)(17,18,23,27)(19,28,31,40)(20,32,34,47)(21,26,22,38)(24,37,41,30)(25,35,43,36)(29,42,33,39)(44,48,46,45);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,37,23,31,22,30)(18,24,38,40,27,41,26,28)(20,33,35,45,34,29,36,48)(25,42,47,44,43,39,32,46);(10,13)(14,16)(17,20,21,35,23,34,22,36)(18,25,38,47,27,43,26,32)(19,29,37,48,31,33,30,45)(24,39,40,46,41,42,28,44);(1,3,5,4)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(20,34)(25,43)(29,33)(32,47)(35,36)(39,42)(44,46)(45,48);(1,4,5,3)(2,6,8,7)(11,15)(14,16)(17,21,23,22)(18,26,27,38)(19,30,31,37)(20,35,34,36)(24,40,41,28)(25,32,43,47)(29,45,33,48)(39,46,42,44)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=gi,d^2=fgi,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ei,d^c=di,e^d=ehi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6466020);