Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6363930 |
| 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 | |
1547 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,9)(5,14)(6,10)(8,11)(12,16)(13,15)(17,18,23,28)(19,29,30,43)(20,25,35,42)(21,27,22,26)(24,36,38,32)(31,45,46,40)(33,48,47,39)(34,44,37,41);(1,3)(2,7)(4,13)(5,11)(6,12)(8,14)(9,15)(10,16)(17,19,23,30)(18,24,28,38)(20,33,35,47)(21,32,22,36)(25,40,42,45)(26,29,27,43)(31,37,46,34)(39,44,48,41);(1,3)(2,8)(4,11)(5,13)(6,12)(7,14)(9,16)(10,15)(19,30)(24,38)(29,43)(31,46)(32,36)(33,47)(39,48)(40,45);(17,20)(18,25)(19,31)(21,34)(22,37)(23,35)(24,39)(26,41)(27,44)(28,42)(29,45)(30,46)(32,47)(33,36)(38,48)(40,43);(1,4,6,5)(2,9,10,14)(3,11,12,13)(7,8,16,15)(17,21,23,22)(18,26,28,27)(19,32,30,36)(20,34,35,37)(24,29,38,43)(25,41,42,44)(31,47,46,33)(39,45,48,40)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ei,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, 6271337);