Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | C2×#546 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1626 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,28,24,29,22,27)(19,30,32,39,33,43,21,25)(20,26,35,40,36,41,37,42)(31,44,45,48,46,47,34,38);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,24,33)(18,25,29,39)(20,31,36,46)(21,22,32,23)(26,38,41,48)(27,30,28,43)(34,37,45,35)(40,47,42,44);(2,6)(5,7)(9,11,12,16)(10,13,14,15)(17,20)(18,26)(19,31)(21,34)(22,37)(23,35)(24,36)(25,38)(27,42)(28,40)(29,41)(30,44)(32,45)(33,46)(39,48)(43,47);(3,8)(5,7)(11,16)(13,15)(17,21,23,19,24,32,22,33)(18,25,28,30,29,39,27,43)(20,34,35,31,36,45,37,46)(26,38,40,44,41,48,42,47);(10,14)(13,15)(17,22,24,23)(18,27,29,28)(19,32,33,21)(20,35,36,37)(25,30,39,43)(26,40,41,42)(31,34,46,45)(38,47,48,44)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=h,d^2=fg,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,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, 6987684);