Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6363286 |
| 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 | #61492 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1496 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)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18,24,29)(19,30,34,46)(20,21,26,39)(22,28,23,27)(25,41,45,33)(31,32,35,48)(36,42,40,38)(37,47,43,44);(17,19,24,34)(18,25,29,45)(20,35,26,31)(21,37,39,43)(22,33,23,41)(27,30,28,46)(32,42,48,38)(36,47,40,44);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,20,24,26)(18,21,29,39)(19,31,34,35)(22,40,23,36)(25,43,45,37)(27,42,28,38)(30,32,46,48)(33,47,41,44);(17,21)(18,26)(19,32)(20,29)(22,38)(23,42)(24,39)(25,44)(27,40)(28,36)(30,35)(31,46)(33,43)(34,48)(37,41)(45,47);(2,7)(6,15)(8,12)(9,16)(17,22,24,23)(18,27,29,28)(19,33,34,41)(20,36,26,40)(21,38,39,42)(25,30,45,46)(31,44,35,47)(32,43,48,37)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=hi,e^2=h,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,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, 6270527);