Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(9,10,13,11)(12,15,16,14)(17,18,21,24)(19,22,25,29)(20,27,28,23)(26,32,31,30)(33,34)(35,39)(36,38)(37,40)(41,43)(42,47)(44,46)(45,48);(2,5)(6,8)(9,11,13,10)(12,15,16,14)(19,25)(22,29)(26,31)(30,32)(34,36)(39,41)(40,44)(47,48);(1,2)(3,6)(4,5)(7,8)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,35,38,43)(34,39,36,41)(37,45,46,42)(40,48,44,47);(1,3,4,7)(2,6,5,8)(9,12,13,16)(10,14,11,15)(17,19,21,25)(18,22,24,29)(20,26,28,31)(23,30,27,32)(33,36,38,34)(35,41,43,39)(37,40,46,44)(42,47,45,48);(17,20,21,28)(18,23,24,27)(19,26,25,31)(22,30,29,32)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,48)(43,45)Polycyclic presentation:
2^9|a^2=fghi,b^2=fhi,c^2=f,d^2=gi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3422544);