Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6131635 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
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 | |
1330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(10,14)(15,16)(17,18,21,24)(19,25,27,22)(20,28,29,23)(26,30,32,31)(33,34)(35,41)(36,40)(37,38)(39,42)(43,47)(44,45)(46,48);(2,5)(6,8)(9,10,13,14)(11,15,12,16)(20,29)(23,28)(26,32)(30,31)(34,38)(35,42)(40,45)(43,48);(9,11,13,12)(10,15,14,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,29)(31,32)(33,34)(35,39)(36,40)(37,38)(41,42)(43,46)(44,45)(47,48);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,15,14,16)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31)(33,35,37,42)(34,39,38,41)(36,43,44,48)(40,46,45,47);(1,3)(2,6)(4,7)(5,8)(9,12,13,11)(10,16,14,15)(17,20)(18,23)(19,26)(21,29)(22,30)(24,28)(25,31)(27,32)(33,36,37,44)(34,40,38,45)(35,43,42,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=fh,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=eg68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6039343);