Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| 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 | |
1294 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,13,15,16)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(33,34,38,36)(35,41,42,39)(37,44,45,40)(43,46,48,47);(1,3)(2,7)(4,6)(5,8)(9,11,12,15)(10,13,14,16)(18,19)(23,25)(24,26)(30,31)(35,42)(37,45)(39,41)(40,44);(1,2,6,5)(3,7,4,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,18)(20,23,28,25)(21,24,29,26)(27,31,32,30)(33,35)(34,39)(36,41)(37,43)(38,42)(40,46)(44,47)(45,48);(1,4,6,3)(2,7,5,8)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,36,38,34)(35,39,42,41)(37,44,45,40)(43,46,48,47);(1,5,6,2)(3,7,4,8)(9,10)(11,13)(12,14)(15,16)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,37,38,45)(34,40,36,44)(35,43,42,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=i,d^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6176841);