Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | |
1388 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)(19,25,33,42)(20,34,37,26)(21,27,24,30)(22,28,32,41)(31,47,48,40)(35,46,39,43)(36,45,38,44);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,33)(18,25,29,42)(20,35,37,39)(21,32,24,22)(26,43,34,46)(27,41,30,28)(31,36,48,38)(40,44,47,45);(10,14)(13,15)(17,20,21,36,23,37,24,38)(18,26,27,44,29,34,30,45)(19,31,22,39,33,48,32,35)(25,40,28,46,42,47,41,43);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,23,24)(18,27,29,30)(19,32,33,22)(20,36,37,38)(25,41,42,28)(26,44,34,45)(31,35,48,39)(40,43,47,46);(2,5)(6,8)(11,16)(13,15)(17,22,24,19,23,32,21,33)(18,28,30,25,29,41,27,42)(20,35,38,48,37,39,36,31)(26,43,45,47,34,46,44,40)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=hi,d^2=fg,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6730605);