Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | #42778 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,25,22,27,21,28)(19,24,36,41,29,40,31,38)(20,32,37,44,34,26,35,42)(30,43,46,47,33,39,45,48);(1,2)(3,8)(4,7)(5,6)(9,10)(11,15)(12,14)(13,16)(17,19,22,29)(18,24,27,40)(20,33,34,30)(21,31,23,36)(25,41,28,38)(26,43,32,39)(35,46,37,45)(42,47,44,48);(9,11,12,16)(10,13,14,15)(18,25)(19,29)(20,34)(21,23)(24,38)(26,44)(27,28)(32,42)(39,48)(40,41)(43,47)(45,46);(1,3,5,4)(2,7,6,8)(11,16)(13,15)(17,20,22,34)(18,26,27,32)(19,30,29,33)(21,35,23,37)(24,39,40,43)(25,42,28,44)(31,45,36,46)(38,47,41,48);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(18,27)(19,29)(20,34)(25,28)(31,36)(35,37)(39,43)(47,48)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=hi,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6980073);