Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470818 |
| 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 | C2×#469 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1562 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)(25,40)(26,39)(27,28)(30,38)(31,43)(34,42)(35,37)(41,48)(44,47);(1,3)(2,6)(4,11)(5,9)(7,15)(8,13)(10,12)(14,16)(17,18,22,24,23,28,21,27)(19,30,33,40,29,25,32,38)(20,34,35,42,36,26,37,39)(31,41,45,47,43,44,46,48);(3,9)(6,13)(10,11)(14,15)(17,19,22,33,23,29,21,32)(18,25,24,38,28,30,27,40)(20,31,35,45,36,43,37,46)(26,41,39,47,34,44,42,48);(1,4,5,12)(2,7,8,16)(3,10,9,11)(6,14,13,15)(17,20)(18,26)(19,31)(21,37)(22,35)(23,36)(24,39)(25,41)(27,42)(28,34)(29,43)(30,44)(32,46)(33,45)(38,47)(40,48);(2,8)(6,13)(7,16)(14,15)(17,21,23,22)(18,27,28,24)(19,32,29,33)(20,35,36,37)(25,40,30,38)(26,39,34,42)(31,45,43,46)(41,47,44,48)Polycyclic presentation:
2^9|b^2=f,c^2=f,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,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, 6389400);