Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6814811 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1609 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,9,11,6)(4,12,13,7)(10,14,16,15)(17,18)(19,28)(20,24)(21,26)(22,27)(23,32)(25,36)(29,30)(31,41)(33,34)(35,40)(37,44)(38,46)(39,47)(42,43)(45,48);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,18)(19,29)(20,33)(21,25)(22,27)(23,32)(24,34)(26,36)(28,30)(31,45)(35,39)(37,44)(38,43)(40,47)(41,48)(42,46);(17,19,22,30)(18,23)(20,31,35,43)(21,32,36,44)(24,38)(25,28)(26,29)(27,37)(33,42)(34,46,47,48)(39,41)(40,45);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(19,30)(23,37)(28,29)(31,43)(32,44)(38,45)(41,42)(46,48);(3,11)(6,9)(10,16)(14,15)(17,20,21,34,22,35,36,47)(18,24,26,33,27,40,25,39)(19,31,32,46,30,43,44,48)(23,38,29,42,37,45,28,41)Polycyclic presentation:
2^9|a^2=f,c^2=gh,d^2=f,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,d^c=di62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6722534);