Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535664 |
| 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 | #25115 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1409 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)(17,18,23,27)(19,28,30,44)(20,33,37,48)(21,26,22,38)(24,31,40,32)(25,35,42,36)(29,41,34,39)(43,45,47,46);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,32,23,30,21,31)(18,24,26,28,27,40,38,44)(20,34,36,45,37,29,35,46)(25,41,33,47,42,39,48,43);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,22,36,23,37,21,35)(18,25,26,33,27,42,38,48)(19,29,32,46,30,34,31,45)(24,39,28,43,40,41,44,47);(1,4,5,3)(2,8,6,7)(9,11)(10,14)(12,15)(13,16)(19,30)(24,40)(28,44)(29,34)(31,32)(39,41)(43,47)(45,46);(11,15)(14,16)(17,21,23,22)(18,26,27,38)(19,31,30,32)(20,35,37,36)(24,28,40,44)(25,33,42,48)(29,45,34,46)(39,43,41,47)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=f,d^2=fgi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6463758);