Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500019 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
2964 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,9)(6,10)(11,12)(14,15)(17,18)(19,27,30,23)(20,31)(21,35)(22,26)(24,33)(25,37)(28,40,32,38)(29,39,36,42)(34,41)(43,47,45,48)(44,46);(1,2)(3,10)(4,7)(5,8)(6,9)(11,15)(12,14)(13,16)(17,19)(18,23)(20,32)(21,36)(22,30)(24,40)(25,42)(26,27)(28,33)(29,37)(31,38)(34,43)(35,39)(41,47)(44,48)(45,46);(1,3,5,9)(2,6,8,10)(4,12,13,11)(7,15,16,14)(17,20,22,33)(18,24,26,31)(19,28,30,32)(21,34,37,46)(23,38,27,40)(25,41,35,44)(29,43,36,45)(39,47,42,48);(1,4,5,13)(2,7,8,16)(3,11,9,12)(6,14,10,15)(20,33)(24,31)(28,32)(34,46)(38,40)(41,44)(43,45)(47,48);(2,8)(6,10)(7,16)(14,15)(17,21,22,37)(18,25,26,35)(19,29,30,36)(20,34,33,46)(23,39,27,42)(24,41,31,44)(28,43,32,45)(38,47,40,48)Polycyclic presentation:
2^9|a^2=gh,c^2=h,d^2=hi,e^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6439803);