Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806671 |
| 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 | |
1398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(3,4)(9,10)(11,15)(12,16)(13,14)(17,18,24,30)(19,25,33,42)(20,26,31,40)(21,27,23,29)(22,36,39,28)(32,47,48,41)(34,45,37,43)(35,46,38,44);(1,2)(3,7)(4,8)(5,6)(17,19,24,33)(18,25,30,42)(20,21,31,23)(22,37,39,34)(26,27,40,29)(28,45,36,43)(32,35,48,38)(41,44,47,46);(1,3,5,4)(2,7,6,8)(10,14)(15,16)(17,20,23,19,24,31,21,33)(18,26,29,25,30,40,27,42)(22,34,38,32,39,37,35,48)(28,43,46,41,36,45,44,47);(1,4,5,3)(2,8,6,7)(9,11,13,12)(10,15,14,16)(17,21,24,23)(18,27,30,29)(19,31,33,20)(22,35,39,38)(25,40,42,26)(28,44,36,46)(32,37,48,34)(41,45,47,43);(2,6)(7,8)(9,12,13,11)(10,16,14,15)(17,22,23,38,24,39,21,35)(18,28,29,46,30,36,27,44)(19,32,31,37,33,48,20,34)(25,41,40,45,42,47,26,43)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fh,d^2=gi,e^2=fgh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6750964);