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 | |
1370 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,8)(3,7,4,5)(9,10,12,14)(11,13,15,16)(18,22)(19,25)(20,28)(30,32)(33,34,37,38)(35,41,43,39)(36,40,44,45)(42,47,48,46);(1,3,6,4)(2,5,8,7)(9,11,12,15)(10,13,14,16)(19,25)(20,28)(23,26)(24,29)(34,38)(36,44)(39,41)(42,48);(3,4)(5,7)(9,11)(10,13)(12,15)(14,16)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30)(33,34)(35,39)(36,45)(37,38)(40,44)(41,43)(42,47)(46,48);(1,4,6,3)(2,7,8,5)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,5,6,7)(2,3,8,4)(9,10)(11,13)(12,14)(15,16)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=h,b^2=fghi,c^2=fi,d^2=f,e^2=fg,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, 6176730);