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)(17,18,22,19)(20,25,27,23)(21,24,30,26)(28,32,29,31)(33,34,37,40)(35,41,43,38)(36,44,45,39)(42,46,48,47);(2,5)(6,8)(9,11,13,10)(12,15,16,14)(18,19)(20,27)(24,26)(28,29)(36,45)(39,44)(42,48)(46,47);(1,2)(3,6)(4,5)(7,8)(17,19,22,18)(20,25,27,23)(21,24,30,26)(28,31,29,32)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,3,4,7)(2,6,5,8)(9,12,13,16)(10,14,11,15)(17,20,22,27)(18,23,19,25)(21,29,30,28)(24,32,26,31)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47);(17,21,22,30)(18,24,19,26)(20,28,27,29)(23,31,25,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,45)(47,48)Polycyclic presentation:
2^9|a^2=fhi,b^2=ghi,c^2=fg,d^2=i,e^2=f,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, 3422222);