Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470037 |
| 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 | C22×C4 | C25 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2800 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)(3,6)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(17,18,23,20)(19,27,31,40)(21,25,34,32)(22,33,37,26)(24,36,28,30)(29,41,43,48)(35,45,46,39)(38,47,42,44);(1,3)(2,6)(4,11)(5,9)(7,15)(8,13)(10,12)(14,16)(17,19)(18,24)(20,28)(21,29)(22,36)(23,31)(25,38)(26,40)(27,33)(30,37)(32,42)(34,43)(35,47)(39,48)(41,45)(44,46);(3,9)(6,13)(10,11)(14,15)(17,20,23,18)(19,28,31,24)(21,32,34,25)(22,26,37,33)(27,36,40,30)(29,42,43,38)(35,39,46,45)(41,47,48,44);(1,4,5,12)(2,7,8,16)(3,10,9,11)(6,14,13,15)(17,21,23,34)(18,25,20,32)(19,29,31,43)(22,35,37,46)(24,38,28,42)(26,39,33,45)(27,41,40,48)(30,44,36,47);(2,8)(6,13)(7,16)(14,15)(21,34)(25,32)(29,43)(35,46)(38,42)(39,45)(41,48)(44,47)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6388823);