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 | |
1402 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)(9,10,12,14)(11,13,15,16)(17,18,21,24)(19,22,27,25)(20,23,28,30)(26,29,32,31)(33,34,38,39)(35,36,42,40)(37,41,46,47)(43,44,45,48);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(20,28)(23,30)(26,32)(29,31)(34,39)(35,42)(41,47)(43,45);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,35,38,42)(34,40,39,36)(37,45,46,43)(41,44,47,48);(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31)(33,36,38,40)(34,35,39,42)(37,44,46,48)(41,43,47,45);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)(33,37,38,46)(34,41,39,47)(35,43,42,45)(36,44,40,48)Polycyclic presentation:
2^9|a^2=fgi,b^2=fi,c^2=f,d^2=f,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, 5393616);