Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6365522 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #61890 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1566 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)(3,6)(4,7)(5,8)(9,12)(10,14)(11,13)(15,16)(17,18,23,27)(19,28,32,40)(20,25,22,36)(21,26,35,43)(24,29,39,31)(30,45,46,42)(33,41,37,44)(34,48,47,38);(1,3)(2,6)(4,11)(5,10)(7,13)(8,14)(9,15)(12,16)(17,19,23,32)(18,24,27,39)(20,31,22,29)(21,34,35,47)(25,40,36,28)(26,42,43,45)(30,37,46,33)(38,41,48,44);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,20,23,22)(18,25,27,36)(19,29,32,31)(21,33,35,37)(24,28,39,40)(26,41,43,44)(30,34,46,47)(38,45,48,42);(2,8)(6,14)(7,9)(13,15)(17,21,23,35)(18,26,27,43)(19,30,32,46)(20,33,22,37)(24,38,39,48)(25,41,36,44)(28,45,40,42)(29,34,31,47);(3,10)(6,14)(11,16)(13,15)(18,27)(24,39)(25,36)(26,43)(28,40)(38,48)(41,44)(42,45)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ei,f^b=fh,g^c=gi,f^d=fh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6272733);