Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6351598 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #62825 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,3)(4,11,12,16)(5,10,15,8)(7,14,9,13)(17,18)(19,29)(20,33)(21,26)(22,27)(23,28)(24,31)(25,35)(30,38)(32,39)(34,42)(36,40)(37,41)(43,46)(44,48)(45,47);(1,3,6,2)(4,9,12,7)(5,8,15,10)(11,14,16,13)(17,19)(18,24)(20,30)(21,36)(22,31)(23,32)(25,38)(26,29)(27,39)(28,40)(33,47)(34,44)(35,46)(37,45)(41,48)(42,43);(1,4)(2,7)(3,9)(5,14)(6,12)(8,16)(10,11)(13,15)(17,20)(18,25)(19,30)(21,35)(22,37)(23,34)(24,38)(26,42)(27,33)(28,41)(29,43)(31,45)(32,44)(36,46)(39,47)(40,48);(20,34)(25,41)(30,44)(33,42)(35,37)(38,48)(43,47)(45,46);(17,21,23,22)(18,26,28,27)(19,31,32,36)(20,35,34,37)(24,39,40,29)(25,42,41,33)(30,45,44,46)(38,47,48,43)Polycyclic presentation:
2^9|a^2=i,b^2=i,e^2=h,f^2=h,g^2=hi,b^a=bf,c^a=cg,f^a=fh,g^a=ghi,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=ghi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258811);