Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632082 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1660 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)(3,5)(4,6)(7,8)(10,14)(11,12)(17,18,23,25,22,28,21,27)(19,24,36,38,30,40,31,41)(20,32,37,44,34,26,35,42)(29,43,46,48,33,39,45,47);(1,3)(2,5)(4,7)(6,8)(9,10)(11,16)(12,15)(13,14)(17,19,22,30)(18,24,28,40)(20,33,34,29)(21,31,23,36)(25,38,27,41)(26,43,32,39)(35,46,37,45)(42,48,44,47);(3,7)(5,8)(9,11,13,12)(10,15,14,16)(18,25)(21,23)(24,38)(26,42)(27,28)(31,36)(32,44)(35,37)(39,47)(40,41)(43,48)(45,46);(3,7)(5,8)(9,12,13,11)(10,15,14,16)(17,20,22,34)(18,26,28,32)(19,29,30,33)(21,35,23,37)(24,39,40,43)(25,42,27,44)(31,45,36,46)(38,47,41,48);(2,6)(5,8)(19,30)(20,34)(24,40)(26,32)(31,36)(35,37)(38,41)(42,44)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=f,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=efh,c^b=ch,d^b=dfhi,e^b=ei,g^c=gi,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6542787);