Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470894 |
| 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 | #37418 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1656 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,30)(20,36)(21,27)(22,28)(23,41)(24,29)(25,34)(26,37)(31,47)(32,44)(33,42)(35,48)(38,43)(39,45)(40,46);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,34,24,35,22,33)(18,25,28,30,29,42,41,48)(20,31,37,40,38,21,32,39)(26,27,36,46,44,47,43,45);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,20,23,37,24,38,22,32)(18,26,28,36,29,44,41,43)(19,31,34,40,35,21,33,39)(25,27,30,46,42,47,48,45);(11,15)(13,16)(17,21,24,31)(18,27,29,47)(19,32,35,37)(20,33,38,34)(22,40,23,39)(25,36,42,43)(26,30,44,48)(28,46,41,45);(10,14)(13,16)(17,22,24,23)(18,28,29,41)(19,33,35,34)(20,37,38,32)(21,39,31,40)(25,30,42,48)(26,43,44,36)(27,45,47,46)Polycyclic presentation:
2^9|b^2=g,c^2=f,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6380094);