Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6366870 |
| 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 | C42 | C25 |
| prime radical | #61996 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1519 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,7)(3,5)(8,15)(9,10)(11,14)(12,16)(17,18,23,28)(19,24,32,39)(20,33,35,40)(21,26,22,27)(25,36,41,34)(29,43,46,47)(30,38,31,42)(37,45,48,44);(1,2,6,10)(3,7,5,9)(4,12,13,15)(8,14,16,11)(17,19)(18,24)(20,29)(21,30)(22,31)(23,32)(25,37)(26,38)(27,42)(28,39)(33,43)(34,44)(35,46)(36,45)(40,47)(41,48);(1,3,6,5)(2,7,10,9)(4,11,13,14)(8,15,16,12)(17,20,23,35)(18,25,28,41)(19,29,32,46)(21,34,22,36)(24,37,39,48)(26,40,27,33)(30,44,31,45)(38,47,42,43);(1,4,6,13)(2,8,10,16)(3,11,5,14)(7,15,9,12)(17,21,23,22)(18,26,28,27)(19,30,32,31)(20,34,35,36)(24,38,39,42)(25,40,41,33)(29,44,46,45)(37,47,48,43);(1,3,6,5)(2,9,10,7)(4,11,13,14)(8,12,16,15)(17,21,23,22)(18,27,28,26)(19,31,32,30)(20,34,35,36)(24,38,39,42)(25,33,41,40)(29,45,46,44)(37,47,48,43)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=hi,d^2=hi,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,d^b=df,e^b=ehi,f^b=fh,g^c=gi,f^d=fh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6274390);