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 | |
1455 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)(19,28,30,45)(20,32,21,25)(22,36,37,26)(24,38,41,31)(29,44,34,43)(33,42,35,47)(39,46,40,48);(9,10)(11,14)(12,13)(15,16)(17,19,22,31,23,30,37,38)(18,24,26,28,27,41,36,45)(20,29,33,46,21,34,35,48)(25,39,42,43,32,40,47,44);(2,5)(6,8)(10,13)(14,16)(17,20)(18,25)(19,29)(21,23)(22,33)(24,39)(26,42)(27,32)(28,43)(30,34)(31,46)(35,37)(36,47)(38,48)(40,41)(44,45);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,21,23,20)(18,25,27,32)(19,29,30,34)(22,35,37,33)(24,40,41,39)(26,42,36,47)(28,44,45,43)(31,46,38,48);(9,11,12,15)(10,14,13,16)(19,30)(20,21)(24,41)(25,32)(28,45)(31,38)(33,35)(42,47)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=dhi,e^b=ei,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, 6402793);