Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1322 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)(10,14)(11,12)(17,18,22,27)(19,24,30,38)(20,31,33,25)(21,26,35,41)(23,28,36,42)(29,43,45,37)(32,46,48,39)(34,47,44,40);(9,10)(11,15)(12,16)(13,14)(17,19,22,30)(18,24,27,38)(20,32,33,48)(21,23,35,36)(25,39,31,46)(26,28,41,42)(29,44,45,34)(37,47,43,40);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,20,23,34,22,33,36,44)(18,25,28,40,27,31,42,47)(19,29,21,32,30,45,35,48)(24,37,26,39,38,43,41,46);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(20,33)(25,31)(29,45)(32,48)(34,44)(37,43)(39,46)(40,47);(10,14)(15,16)(17,21,23,30,22,35,36,19)(18,26,28,38,27,41,42,24)(20,32,34,45,33,48,44,29)(25,39,40,43,31,46,47,37)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fh,d^2=gi,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6729739);