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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1370 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18,23,29)(19,27,32,25)(20,34,37,45)(21,39,33,42)(22,30,24,28)(26,38,44,36)(31,47,48,43)(35,46,40,41);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,32)(18,25,29,27)(20,35,37,40)(21,24,33,22)(26,43,44,47)(28,39,30,42)(31,38,48,36)(34,41,45,46);(10,14)(11,16)(17,20,24,38,23,37,22,36)(18,26,30,45,29,44,28,34)(19,31,21,40,32,48,33,35)(25,41,39,47,27,46,42,43);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,24,32,23,33,22,19)(18,27,30,42,29,25,28,39)(20,35,38,31,37,40,36,48)(26,41,45,47,44,46,34,43);(2,5)(6,8)(10,14)(13,15)(17,22,23,24)(18,28,29,30)(19,21,32,33)(20,36,37,38)(25,39,27,42)(26,34,44,45)(31,40,48,35)(41,47,46,43)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=h,d^2=fg,e^2=i,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, 7510094);