Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| 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 | |
1317 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)(4,8)(5,6)(10,13)(11,15)(17,18)(19,29)(20,25)(21,36)(22,26)(23,28)(24,31)(27,38)(30,45)(32,41)(33,42)(34,40)(35,43)(37,44)(39,47)(46,48);(9,10)(11,14)(12,13)(15,16)(17,19,22,31)(18,24,26,29)(20,32,33,34)(21,23,38,37)(25,40,42,41)(27,28,36,44)(30,46,47,35)(39,48,45,43);(2,6)(7,8)(10,13)(14,16)(17,20,23,35,22,33,37,46)(18,25,28,43,26,42,44,48)(19,30,21,32,31,47,38,34)(24,39,27,40,29,45,36,41);(1,3,5,4)(2,7,6,8)(9,11,12,15)(10,14,13,16)(18,26)(20,33)(24,29)(27,36)(28,44)(30,47)(32,34)(35,46);(1,4,5,3)(2,8,6,7)(17,21,37,19,22,38,23,31)(18,27,44,24,26,36,28,29)(20,34,46,47,33,32,35,30)(25,41,48,45,42,40,43,39)Polycyclic presentation:
2^9|b^2=i,c^2=fhi,d^2=fgi,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=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6753471);