Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 | #37700 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1514 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)(3,6)(4,5)(7,8)(17,18)(19,27)(20,31)(21,23)(22,36)(24,30)(25,33)(26,35)(28,43)(29,41)(32,42)(34,40)(37,46)(38,48)(39,45)(44,47);(1,3,7,4)(2,5,8,6)(9,10)(11,15)(12,14)(13,16)(18,23)(24,37)(25,31)(26,36)(27,41)(38,39)(40,47)(42,43);(1,4,7,3)(2,5,8,6)(9,11)(10,13)(12,16)(14,15)(17,19,22,30,21,29,35,46)(18,24,26,27,23,37,36,41)(20,32,34,48,33,28,44,45)(25,39,40,43,31,38,47,42);(1,5,7,6)(2,3,8,4)(11,16)(13,15)(17,20,21,33)(18,25,23,31)(19,28,29,32)(22,34,35,44)(24,38,37,39)(26,40,36,47)(27,42,41,43)(30,45,46,48);(1,6,7,5)(2,3,8,4)(10,14)(13,15)(20,33)(25,31)(28,32)(34,44)(38,39)(40,47)(42,43)(45,48)Polycyclic presentation:
2^9|b^2=fi,c^2=fgi,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,d^c=dfh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6822962);