Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438943 |
| 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 | #2431 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1543 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)(17,18,24,30,23,28,22,29)(19,31,36,25,35,44,34,43)(20,37,39,46,38,26,21,27)(32,45,33,47,40,42,48,41);(2,6)(7,8)(17,19,23,35)(18,25,28,43)(20,33,38,48)(21,40,39,32)(22,36,24,34)(26,45,37,42)(27,41,46,47)(29,44,30,31);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,24,39,23,38,22,21)(18,26,30,27,28,37,29,46)(19,32,36,33,35,40,34,48)(25,41,44,45,43,47,31,42);(1,4,5,3)(2,8,6,7)(17,21,22,38,23,39,24,20)(18,27,29,26,28,46,30,37)(19,33,34,32,35,48,36,40)(25,42,31,47,43,45,44,41);(9,11,12,15)(10,14,13,16)(18,28)(25,43)(26,37)(27,46)(29,30)(31,44)(41,47)(42,45)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=fi,d^2=f,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6366380);