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 | #45243 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1549 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,29)(20,33)(21,24)(22,40)(23,26)(25,35)(27,32)(28,30)(31,46)(34,44)(36,41)(37,45)(38,43)(39,42)(47,48);(2,5)(3,8)(9,11)(10,13)(12,16)(14,15)(17,19,30,39,23,32,22,21)(18,24,40,27,26,42,28,29)(20,34,47,38,35,31,37,36)(25,43,48,44,33,41,45,46);(1,3,4,8)(2,6,5,7)(10,14)(11,16)(17,20,23,35)(18,25,26,33)(19,31,32,34)(21,38,39,36)(22,37,30,47)(24,41,42,43)(27,46,29,44)(28,45,40,48);(2,5)(6,7)(10,14)(11,16)(18,26)(19,32)(20,35)(21,39)(28,40)(37,47)(41,43)(44,46);(2,5)(6,7)(10,14)(13,15)(17,21,22,32,23,39,30,19)(18,27,28,24,26,29,40,42)(20,36,37,31,35,38,47,34)(25,44,45,43,33,46,48,41)Polycyclic presentation:
2^9|b^2=fg,c^2=ghi,e^2=fgi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7508130);