Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6368846 |
| 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 | #62609 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,7,12)(3,13,14,9)(4,10,15,16)(5,11,6,8)(17,18)(19,25)(20,31)(21,26)(22,35)(23,29)(24,33)(27,34)(28,43)(30,38)(32,42)(36,40)(37,47)(39,45)(41,46)(44,48);(2,8)(4,9)(5,6)(10,16)(11,12)(13,15)(17,19,21,29)(18,23,26,25)(20,28,32,44)(22,30,34,46)(24,37,40,39)(27,38,35,41)(31,48,42,43)(33,45,36,47);(1,3,7,14)(2,9,12,13)(4,11,15,8)(5,16,6,10)(17,20,21,32)(18,24,26,40)(19,28,29,44)(22,36,34,33)(23,37,25,39)(27,42,35,31)(30,47,46,45)(38,43,41,48);(1,4)(2,10)(3,11)(5,9)(6,13)(7,15)(8,14)(12,16)(17,19)(18,25)(20,28)(21,29)(22,30)(23,26)(24,39)(27,41)(31,43)(32,44)(33,45)(34,46)(35,38)(36,47)(37,40)(42,48);(1,5,7,6)(2,8,12,11)(3,10,14,16)(4,9,15,13)(19,29)(23,25)(28,44)(30,46)(37,39)(38,41)(43,48)(45,47)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=hi,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,e^d=ei,f^d=fh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6276150);