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)(17,18,23,28)(19,30,32,25)(20,34,37,45)(21,39,33,42)(22,29,24,27)(26,36,46,38)(31,43,47,48)(35,41,40,44);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,32)(18,25,28,30)(20,35,37,40)(21,24,33,22)(26,43,46,48)(27,39,29,42)(31,38,47,36)(34,44,45,41);(1,3,5,4)(2,7,6,8)(17,20,22,36,23,37,24,38)(18,26,27,45,28,46,29,34)(19,31,33,35,32,47,21,40)(25,41,42,43,30,44,39,48);(1,3,5,4)(2,7,6,8)(11,15)(13,16)(17,21,22,19,23,33,24,32)(18,25,27,42,28,30,29,39)(20,35,36,47,37,40,38,31)(26,44,45,48,46,41,34,43);(1,4,5,3)(2,7,6,8)(17,22,23,24)(18,27,28,29)(19,21,32,33)(20,36,37,38)(25,39,30,42)(26,45,46,34)(31,40,47,35)(41,48,44,43)Polycyclic presentation:
2^9|a^2=fgh,b^2=i,c^2=fg,d^2=fg,e^2=f,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=cgi,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, 7501395);