Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6354774 |
| 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 | |
1526 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,15)(12,16)(17,18)(19,29)(20,34)(21,26)(22,27)(23,28)(24,33)(25,36)(30,44)(31,40)(32,39)(35,42)(37,41)(38,47)(43,46)(45,48);(1,3,5,13,6,12,4,11)(2,7,8,10,9,15,14,16)(17,19,21,33,23,31,22,32)(18,24,27,29,28,39,26,40)(20,30,36,46,35,47,37,45)(25,38,34,43,41,44,42,48);(1,2)(3,7)(4,14)(5,8)(6,9)(10,13)(11,16)(12,15)(17,20)(18,25)(19,30)(21,36)(22,37)(23,35)(24,38)(26,42)(27,34)(28,41)(29,43)(31,47)(32,45)(33,46)(39,44)(40,48);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,10,15,16)(19,31)(20,35)(24,39)(25,41)(29,40)(32,33)(34,42)(36,37);(3,12)(7,15)(10,16)(11,13)(17,21,23,22)(18,26,28,27)(19,32,31,33)(20,36,35,37)(24,29,39,40)(25,42,41,34)(30,45,47,46)(38,43,44,48)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,f^a=fh,g^a=gi,d^b=di,e^b=eh,d^c=dh,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72306948, #72306952, #72306958, … |
| maximal normal subgroup of | #72306948, #72306952, #72306958, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6262026);