Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6358057 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #62546 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,8)(3,13)(5,6)(10,14)(11,12)(15,16)(17,18,22,26)(19,27,28,40)(20,24,31,38)(21,33,34,25)(23,35,36,30)(29,41,42,48)(32,45,44,39)(37,46,47,43);(1,2,5,8,7,11,6,12)(3,4,10,15,14,9,13,16)(17,19,21,30,22,28,34,35)(18,23,25,27,26,36,33,40)(20,29,32,43,31,42,44,46)(24,37,39,41,38,47,45,48);(1,3,6,13,7,14,5,10)(2,9,12,15,11,4,8,16)(19,28)(20,31)(23,36)(24,38)(27,40)(30,35)(32,44)(39,45);(1,4,7,9)(2,10,11,13)(3,8,14,12)(5,15,6,16)(17,20,22,31)(18,24,26,38)(19,29,28,42)(21,32,34,44)(23,37,36,47)(25,39,33,45)(27,41,40,48)(30,43,35,46);(2,11)(4,9)(8,12)(15,16)(20,31)(24,38)(29,42)(32,44)(37,47)(39,45)(41,48)(43,46)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=g,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,e^b=ei,d^c=dhi,e^d=eh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6265278);