Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6368046 |
| 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 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1490 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,9)(3,10,12,15)(4,8,6,13)(7,14,16,11)(17,18,24,29)(19,30,35,44)(20,26,37,21)(22,28,23,27)(25,33,45,34)(31,32,46,48)(36,39,40,41)(38,47,42,43);(1,2)(3,7)(4,13)(5,9)(6,8)(10,11)(12,16)(14,15)(17,19,24,35)(18,25,29,45)(20,31,37,46)(21,38,26,42)(22,34,23,33)(27,30,28,44)(32,41,48,39)(36,43,40,47);(1,3,5,12)(2,7,9,16)(4,11,6,14)(8,15,13,10)(17,20)(18,26)(19,31)(21,29)(22,40)(23,36)(24,37)(25,42)(27,39)(28,41)(30,32)(33,43)(34,47)(35,46)(38,45)(44,48);(1,4,5,6)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,21,24,26)(18,20,29,37)(19,32,35,48)(22,39,23,41)(25,43,45,47)(27,36,28,40)(30,46,44,31)(33,38,34,42);(2,9)(7,16)(8,13)(10,15)(17,22,24,23)(18,27,29,28)(19,33,35,34)(20,36,37,40)(21,39,26,41)(25,44,45,30)(31,47,46,43)(32,38,48,42)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=i,d^2=hi,e^2=h,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,e^c=eh,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, 6275685);