Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | #35771 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1752 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,18,23,29)(19,25,33,43)(20,34,37,26)(21,27,24,30)(22,39,31,28)(32,42,40,47)(35,46,38,44)(36,48,41,45);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,33)(18,25,29,43)(20,35,37,38)(21,32,24,40)(22,36,31,41)(26,44,34,46)(27,42,30,47)(28,45,39,48);(10,14)(13,15)(17,20,23,37)(18,26,29,34)(19,31,33,22)(21,36,24,41)(25,39,43,28)(27,45,30,48)(32,38,40,35)(42,46,47,44);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,23,24)(18,27,29,30)(19,32,33,40)(20,36,37,41)(22,35,31,38)(25,42,43,47)(26,45,34,48)(28,44,39,46);(2,5)(6,8)(11,16)(13,15)(17,22,23,31)(18,28,29,39)(19,20,33,37)(21,38,24,35)(25,26,43,34)(27,46,30,44)(32,41,40,36)(42,48,47,45)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=fg,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6606301);