Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,7,4,8)(9,10,12,14)(11,13,15,16)(17,18)(19,25)(20,32)(21,38)(22,28)(23,40)(24,29)(26,36)(27,31)(30,43)(33,41)(34,45)(35,44)(37,47)(39,42)(46,48);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,24,31)(18,25,29,27)(20,33,37,34)(21,23,39,22)(26,43,44,48)(28,38,40,42)(30,35,46,36)(32,41,47,45);(1,3,5,4)(2,7,6,8)(17,20,23,36,24,37,22,35)(18,26,28,32,29,44,40,47)(19,30,21,33,31,46,39,34)(25,41,42,43,27,45,38,48);(1,3,5,4)(2,7,6,8)(11,15)(13,16)(17,21,23,31,24,39,22,19)(18,27,28,38,29,25,40,42)(20,34,36,30,37,33,35,46)(26,41,32,43,44,45,47,48);(1,4,5,3)(2,7,6,8)(17,22,24,23)(18,28,29,40)(19,21,31,39)(20,35,37,36)(25,38,27,42)(26,32,44,47)(30,33,46,34)(41,48,45,43)Polycyclic presentation:
2^9|a^2=fghi,b^2=i,c^2=fg,d^2=fg,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7501388);