Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6354237 |
| 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 | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1578 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)(3,11)(4,9)(5,8)(6,10)(7,12)(13,15)(14,16)(17,18)(19,28)(20,25)(21,26)(22,36)(23,27)(24,32)(29,41)(30,38)(31,42)(33,39)(34,40)(35,46)(37,44)(43,47)(45,48);(17,19,22,32,23,31,21,30)(18,24,26,28,27,38,36,42)(20,29,35,44,34,45,33,43)(25,37,39,41,40,47,46,48);(1,3)(2,7)(4,12)(5,14)(6,13)(8,11)(9,15)(10,16)(17,20)(18,25)(19,29)(21,33)(22,35)(23,34)(24,37)(26,39)(27,40)(28,41)(30,43)(31,45)(32,44)(36,46)(38,47)(42,48);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,11,16,15)(17,21,23,22)(18,26,27,36)(19,30,31,32)(20,33,34,35)(24,28,38,42)(25,39,40,46)(29,43,45,44)(37,41,47,48);(1,4,6,5)(2,9,10,8)(3,12,13,14)(7,15,16,11)(19,31)(20,34)(24,38)(25,40)(28,42)(30,32)(33,35)(39,46)Polycyclic presentation:
2^9|b^2=f,d^2=hi,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,e^b=eh,e^c=eh,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72306650, #72306653, #72306654, … |
| maximal normal subgroup of | #72306650, #72306653, #72306654, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6261461);