Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle, prime radical | C23 | C24×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1080 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,11)(4,8)(5,9)(6,10)(7,13)(12,16)(14,15)(17,18)(19,22)(20,27)(21,24)(23,29)(25,32)(26,31)(28,30)(33,34,40,36,39,43,37,47)(35,44,46,45,42,41,48,38);(1,3,6,13)(2,7,10,11)(4,12,5,14)(8,15,9,16)(17,19,21,26)(18,22,24,31)(20,28,29,25)(23,32,27,30)(33,35)(34,41)(36,38)(37,48)(39,42)(40,46)(43,44)(45,47);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(17,20,21,29)(18,23,24,27)(19,25,26,28)(22,30,31,32)(33,36,40,43,39,47,37,34)(35,45,46,41,42,38,48,44);(1,5,6,4)(2,9,10,8)(3,12,13,14)(7,15,11,16)(18,24)(22,31)(23,27)(30,32)(33,37,39,40)(34,36,43,47)(35,46,42,48)(38,41,45,44);(2,10)(7,11)(8,9)(15,16)(33,38,40,44,39,45,37,41)(34,42,36,48,43,35,47,46)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=fgh,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df,e^b=ei,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6338498);