Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535823 |
| 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 | #2540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1543 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,7,4,6)(17,18)(19,24)(20,31)(21,26)(22,37)(23,27)(25,35)(28,44)(29,39)(30,41)(32,46)(33,47)(34,43)(36,40)(38,45)(42,48);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,30)(18,24,27,41)(20,32,33,28)(21,36,22,29)(25,42,43,38)(26,40,37,39)(31,46,47,44)(34,45,35,48);(10,13)(14,16)(17,20,21,34,23,33,22,35)(18,25,37,47,27,43,26,31)(19,28,36,48,30,32,29,45)(24,38,39,46,41,42,40,44);(1,3,5,4)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(20,33)(25,43)(28,32)(31,47)(34,35)(38,42)(44,46)(45,48);(1,4,5,3)(2,6,8,7)(11,15)(14,16)(17,21,23,22)(18,26,27,37)(19,29,30,36)(20,34,33,35)(24,39,41,40)(25,31,43,47)(28,45,32,48)(38,46,42,44)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=gi,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ei,d^c=di,e^d=ehi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6466942);