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 |
| prime radical | #62823 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
2868 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
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)(21,34)(22,26)(24,33)(25,35)(28,39,42,36)(29,37,43,40)(32,38)(41,46,48,47)(44,45);(1,2)(3,10)(4,7)(5,8)(6,9)(11,15)(12,14)(13,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,36)(25,37)(26,27)(31,39)(32,41)(33,42)(34,40)(35,43)(38,46)(44,47)(45,48);(1,3,5,9)(2,6,8,10)(4,12,13,11)(7,15,16,14)(17,20)(18,24)(19,28)(21,32)(22,33)(23,36)(25,38)(26,31)(27,39)(29,41)(30,42)(34,44)(35,45)(37,46)(40,47)(43,48);(1,4,5,13)(2,7,8,16)(3,11,9,12)(6,14,10,15);(2,8)(6,10)(7,16)(14,15)(17,21,22,35)(18,25,26,34)(19,29,30,43)(20,32,33,45)(23,37,27,40)(24,38,31,44)(28,41,42,48)(36,46,39,47)Polycyclic presentation:
2^9|a^2=ghi,c^2=h,d^2=h,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, 6439802);