Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,16)(12,15)(13,14)(17,18,22,27)(19,24,32,42)(20,33,36,25)(21,26,23,28)(29,43,34,39)(30,48,47,40)(31,41,38,46)(35,45,37,44);(1,3,5,4)(2,6,8,7)(9,10)(11,16)(12,15)(13,14)(17,19,22,32)(18,24,27,42)(20,34,36,29)(21,31,23,38)(25,43,33,39)(26,41,28,46)(30,37,47,35)(40,45,48,44);(1,3,5,4)(2,7,8,6)(10,14)(15,16)(17,20,22,36)(18,25,27,33)(19,29,32,34)(21,37,23,35)(24,39,42,43)(26,45,28,44)(30,38,47,31)(40,46,48,41);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,30)(21,35)(22,36)(23,37)(24,40)(26,44)(27,33)(28,45)(29,31)(32,47)(34,38)(39,41)(42,48)(43,46);(1,4,5,3)(2,7,8,6)(9,12,13,11)(10,15,14,16)(17,21,22,23)(18,26,27,28)(19,31,32,38)(20,35,36,37)(24,41,42,46)(25,44,33,45)(29,47,34,30)(39,48,43,40)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=fi,d^2=fgi,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7496110);