Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438991 |
| 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×#540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1524 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)(9,10)(11,14)(12,13)(15,16)(17,18,22,27,23,28,37,39)(19,29,21,26,31,44,36,24)(20,32,35,47,34,25,40,43)(30,45,33,38,46,41,48,42);(2,6)(7,8)(17,19,23,31)(18,24,28,26)(20,33,34,48)(21,22,36,37)(25,41,32,45)(27,44,39,29)(30,40,46,35)(38,47,42,43);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,22,35,23,34,37,40)(18,25,27,43,28,32,39,47)(19,30,21,33,31,46,36,48)(24,38,29,41,26,42,44,45);(1,4,5,3)(2,8,6,7)(17,21,22,31,23,36,37,19)(18,26,27,44,28,24,39,29)(20,33,35,46,34,48,40,30)(25,42,43,45,32,38,47,41);(9,11,12,15)(10,14,13,16)(20,34)(25,32)(30,46)(33,48)(35,40)(38,42)(41,45)(43,47)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=f,d^2=f,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6366464);