Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | |
1304 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)(5,8)(11,15)(13,16)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29)(33,34,37,38)(35,39,43,42)(36,44,45,40)(41,47,48,46);(1,2,4,6)(3,5,7,8)(20,28)(23,30)(26,32)(29,31)(34,38)(39,42)(40,44)(46,47);(1,2)(3,5)(4,6)(7,8)(9,10,12,14)(11,13,15,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,35)(34,39)(36,41)(37,43)(38,42)(40,46)(44,47)(45,48);(9,11)(10,13)(12,15)(14,16)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(33,36,37,45)(34,40,38,44)(35,41,43,48)(39,46,42,47);(1,3)(2,5)(4,7)(6,8)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)(33,34)(35,42)(36,40)(37,38)(39,43)(41,47)(44,45)(46,48)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=fgi,d^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5399760);