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×#550 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1617 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,24,28,23,30,22,29)(19,26,32,25,20,34,33,42)(21,36,38,44,40,27,39,45)(31,46,47,48,35,43,37,41);(9,10)(11,15)(12,14)(13,16)(17,19,23,20)(18,25,30,42)(21,37,40,47)(22,32,24,33)(26,29,34,28)(27,43,36,46)(31,38,35,39)(41,45,48,44);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,23,19)(18,26,30,34)(21,35,40,31)(22,32,24,33)(25,29,42,28)(27,43,36,46)(37,39,47,38)(41,44,48,45);(11,16)(13,15)(17,21,24,38,23,40,22,39)(18,27,28,45,30,36,29,44)(19,31,32,47,20,35,33,37)(25,41,34,46,42,48,26,43);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,22,23,24)(18,28,30,29)(19,32,20,33)(21,38,40,39)(25,26,42,34)(27,44,36,45)(31,37,35,47)(41,46,48,43)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=gh,d^2=fi,e^2=g,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,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, 6998357);