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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1250 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)(19,28)(20,33)(21,37)(22,27)(23,26)(24,32)(25,36)(29,40)(30,41)(31,42)(34,44)(35,45)(38,46)(39,43)(47,48);(1,3,5,4)(2,6,8,7)(9,10)(11,16)(12,15)(13,14)(17,19,22,32)(18,24,27,28)(20,34,36,29)(21,31,23,39)(25,44,33,40)(26,42,37,43)(30,38,48,35)(41,45,47,46);(1,3,5,4)(2,7,8,6)(10,14)(15,16)(17,20,22,36)(18,25,27,33)(19,29,32,34)(21,38,23,35)(24,40,28,44)(26,46,37,45)(30,39,48,31)(41,42,47,43);(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)(23,38)(24,41)(26,45)(27,33)(28,47)(29,31)(32,48)(34,39)(37,46)(40,43)(42,44);(1,4,5,3)(2,7,8,6)(9,12,13,11)(10,15,14,16)(17,21,22,23)(18,26,27,37)(19,31,32,39)(20,35,36,38)(24,42,28,43)(25,45,33,46)(29,48,34,30)(40,41,44,47)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=f,d^2=fg,e^2=fgi,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, 7496097);