Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470003 |
| 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 | #2619 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1567 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)(11,15)(12,16)(13,14)(17,18,22,28,23,27,37,43)(19,29,32,40,33,44,36,24)(20,25,34,41,21,26,35,42)(30,45,47,39,31,46,48,38);(9,10)(11,16)(12,15)(13,14)(17,19,23,33)(18,24,27,40)(20,31,21,30)(22,36,37,32)(25,39,26,38)(28,44,43,29)(34,47,35,48)(41,45,42,46);(2,5)(6,8)(10,14)(15,16)(17,20)(18,25)(19,30)(21,23)(22,34)(24,38)(26,27)(28,41)(29,45)(31,33)(32,47)(35,37)(36,48)(39,40)(42,43)(44,46);(9,11,13,12)(10,15,14,16)(17,21,23,20)(18,26,27,25)(19,31,33,30)(22,35,37,34)(24,39,40,38)(28,42,43,41)(29,46,44,45)(32,48,36,47);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,14,16)(18,27)(20,21)(24,40)(28,43)(29,44)(30,31)(34,35)(47,48)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6392514);