Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437418 |
| 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 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,8,4,6)(17,18,23,27)(19,28,31,39)(20,32,34,25)(21,26,22,36)(24,37,40,30)(29,43,45,42)(33,47,35,41)(38,44,46,48);(9,10)(11,14)(12,13)(15,16)(17,19,23,31)(18,24,27,40)(20,29,34,45)(21,30,22,37)(25,38,32,46)(26,39,36,28)(33,44,35,48)(41,43,47,42);(1,3,5,4)(2,6,7,8)(10,13)(14,16)(17,20,23,34)(18,25,27,32)(19,29,31,45)(21,33,22,35)(24,38,40,46)(26,41,36,47)(28,42,39,43)(30,44,37,48);(10,13)(14,16)(17,21,23,22)(18,26,27,36)(19,30,31,37)(20,33,34,35)(24,39,40,28)(25,41,32,47)(29,44,45,48)(38,43,46,42);(1,4,5,3)(2,6,7,8)(9,11,12,15)(10,14,13,16)(20,34)(25,32)(29,45)(33,35)(38,46)(41,47)(42,43)(44,48)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=g,d^2=i,e^2=ghi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6376630);