Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500695 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1241 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,4)(5,7)(6,8)(9,10,12,13)(11,15,16,14)(17,18)(19,24)(20,25)(21,34)(22,26)(23,27)(28,38)(29,47)(30,40)(31,41)(32,39)(33,42)(35,44)(36,43)(37,45)(46,48);(1,3)(2,6)(4,7)(5,8)(17,19,23,31)(18,24,27,41)(20,32,21,29)(22,36,37,30)(25,39,34,47)(26,43,45,40)(28,35,46,33)(38,44,48,42);(10,13)(14,15)(17,20,23,21)(18,25,27,34)(19,28,31,46)(22,35,37,33)(24,38,41,48)(26,44,45,42)(29,36,32,30)(39,40,47,43);(2,7)(3,8)(9,11,12,16)(10,14,13,15)(17,21,23,20)(18,25,27,34)(19,29,31,32)(22,35,37,33)(24,39,41,47)(26,42,45,44)(28,36,46,30)(38,40,48,43);(1,4,5,6)(2,3,7,8)(20,21)(25,34)(28,46)(29,32)(33,35)(38,48)(39,47)(42,44)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,d^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,h^b=hi,e^c=ei,h^c=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7418164);