Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6357480 |
| 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 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1884 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,7,10)(3,5,9,6)(4,8,13,16)(11,12,15,14)(17,18)(19,24)(20,32)(21,26)(22,36)(23,27)(25,34)(28,41)(29,38)(30,39)(31,45)(33,40)(35,46)(37,44)(42,48)(43,47);(1,3,6,2,7,9,5,10)(4,11,14,8,13,15,12,16)(17,19)(18,24)(20,28)(21,30)(22,31)(23,29)(25,37)(26,39)(27,38)(32,41)(33,43)(34,44)(35,42)(36,45)(40,47)(46,48);(1,4,7,13)(2,8,10,16)(3,11,9,15)(5,12,6,14)(17,20,22,34,23,35,21,33)(18,25,26,32,27,40,36,46)(19,28,31,44,29,42,30,43)(24,37,39,41,38,47,45,48);(1,5,7,6)(2,9,10,3)(4,12,13,14)(8,15,16,11)(19,29)(24,38)(28,42)(30,31)(37,47)(39,45)(41,48)(43,44);(4,13)(8,16)(11,15)(12,14)(17,21,23,22)(18,26,27,36)(19,30,29,31)(20,33,35,34)(24,39,38,45)(25,32,40,46)(28,43,42,44)(37,41,47,48)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=gh,d^2=h,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=dhi,e^c=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264711);