Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470421 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60568 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1668 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)(18,24)(19,29)(21,22)(26,39)(27,28)(31,43)(32,33)(34,42)(35,37)(38,41)(45,46)(47,48);(1,3)(2,6)(4,11)(5,9)(7,15)(8,13)(10,12)(14,16)(17,18,22,24,23,28,21,27)(19,30,32,41,33,25,29,38)(20,34,35,42,36,26,37,39)(31,40,45,47,46,44,43,48);(3,9)(6,13)(10,11)(14,15)(17,19)(18,25)(20,31)(21,29)(22,32)(23,33)(24,38)(26,40)(27,41)(28,30)(34,44)(35,45)(36,46)(37,43)(39,47)(42,48);(1,4,5,12)(2,7,8,16)(3,10,9,11)(6,14,13,15)(17,20,23,36)(18,26,28,34)(19,31,33,46)(21,37,22,35)(24,39,27,42)(25,40,30,44)(29,43,32,45)(38,47,41,48);(2,8)(6,13)(7,16)(14,15)(17,21,23,22)(18,27,28,24)(19,29,33,32)(20,35,36,37)(25,41,30,38)(26,39,34,42)(31,45,46,43)(40,47,44,48)Polycyclic presentation:
2^9|b^2=f,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6389101);