Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470395 |
| 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 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1447 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,6,7,8)(11,15)(14,16)(17,18,23,27)(19,28,32,43)(20,33,21,25)(22,36,37,26)(24,38,40,31)(29,44,30,42)(34,41,35,48)(39,45,47,46);(9,10)(11,14)(12,13)(15,16)(17,19,22,31,23,32,37,38)(18,24,26,28,27,40,36,43)(20,29,34,45,21,30,35,46)(25,39,41,42,33,47,48,44);(2,5)(6,8)(10,13)(14,16)(17,20)(18,25)(19,29)(21,23)(22,34)(24,39)(26,41)(27,33)(28,42)(30,32)(31,45)(35,37)(36,48)(38,46)(40,47)(43,44);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,21,23,20)(18,25,27,33)(19,30,32,29)(22,35,37,34)(24,39,40,47)(26,41,36,48)(28,42,43,44)(31,46,38,45);(9,11,12,15)(10,14,13,16)(20,21)(25,33)(29,30)(34,35)(39,47)(41,48)(42,44)(45,46)Polycyclic presentation:
2^9|a^2=g,b^2=f,d^2=g,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di,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, 6402767);