Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6362318 |
| 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 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1594 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)(4,12)(5,9)(6,10)(8,11)(13,15)(14,16)(17,18,23,28)(19,29,30,43)(20,25,35,42)(21,27,22,26)(24,36,38,32)(31,45,46,40)(33,48,47,39)(34,44,37,41);(1,3,6,5)(2,7,10,9)(4,11,13,14)(8,15,16,12)(17,19)(18,24)(20,33)(21,32)(22,36)(23,30)(25,40)(26,29)(27,43)(28,38)(31,34)(35,47)(37,46)(39,41)(42,45)(44,48);(1,4)(2,8)(3,11)(5,14)(6,13)(7,15)(9,12)(10,16)(19,30)(24,38)(29,43)(31,46)(32,36)(33,47)(39,48)(40,45);(17,20,23,35)(18,25,28,42)(19,31,30,46)(21,34,22,37)(24,39,38,48)(26,41,27,44)(29,45,43,40)(32,47,36,33);(1,3,6,5)(2,9,10,7)(4,11,13,14)(8,12,16,15)(17,21,23,22)(18,26,28,27)(19,32,30,36)(20,34,35,37)(24,29,38,43)(25,41,42,44)(31,47,46,33)(39,45,48,40)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,c^b=ch,d^b=df,f^b=fh,g^c=gi,f^d=fh65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6269651);