Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6365146 |
| 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 | #61667 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1559 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,9)(3,6,10,14)(4,7,12,8)(11,13,16,15)(18,24)(21,22)(25,37)(26,38)(27,28)(30,35)(32,41)(33,36)(39,47)(40,42)(43,48)(44,45);(1,3)(2,6)(4,11)(5,10)(7,13)(8,15)(9,14)(12,16)(17,18,23,28)(19,25,31,40)(20,32,34,38)(21,24,22,27)(26,36,41,33)(29,43,46,47)(30,42,35,37)(39,44,48,45);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,19,23,31)(18,25,28,40)(20,29,34,46)(21,35,22,30)(24,37,27,42)(26,39,41,48)(32,43,38,47)(33,45,36,44);(17,20)(18,26)(19,29)(21,33)(22,36)(23,34)(24,38)(25,39)(27,32)(28,41)(30,44)(31,46)(35,45)(37,47)(40,48)(42,43);(3,10)(6,14)(11,16)(13,15)(17,21,23,22)(18,27,28,24)(19,30,31,35)(20,33,34,36)(25,37,40,42)(26,32,41,38)(29,44,46,45)(39,47,48,43)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=ehi,f^b=fh,e^c=eh,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, 6272370);