Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6359920 |
| 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 | #61887 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1500 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)(3,7)(4,11)(5,14)(6,9)(8,10)(12,16)(13,15)(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);(1,3,6,5)(2,7,9,14)(4,12,13,10)(8,11,16,15)(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);(1,4)(2,8)(3,10)(5,12)(6,13)(7,15)(9,16)(11,14)(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);(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);(1,3,6,5)(2,7,9,14)(4,10,13,12)(8,15,16,11)(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=hi,c^2=h,d^2=h,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,c^b=chi,d^b=df,f^b=fh,g^c=gi,f^d=fh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6267355);