Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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 | |
1308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,6)(4,8)(5,7)(11,15)(14,16)(17,18,22,19)(20,25,27,23)(21,26,29,24)(28,30,32,31)(33,34)(35,38)(36,41)(37,40)(39,44)(42,45)(43,47)(46,48);(1,3,5,8)(2,4,7,6)(10,13)(14,16)(20,27)(23,25)(28,32)(30,31)(34,35)(37,45)(39,41)(43,48);(1,3,5,8)(2,6,7,4)(9,10)(11,14)(12,13)(15,16)(17,19,22,18)(20,25,27,23)(21,26,29,24)(28,31,32,30)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,3)(2,4)(5,8)(6,7)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,36,38,44)(34,39,35,41)(37,43,45,48)(40,46,42,47);(1,4)(2,3)(5,6)(7,8)(9,11,12,15)(10,14,13,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=fi,b^2=fg,c^2=i,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5941614);