Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| 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 | |
1368 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)(3,4)(5,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,20)(19,25,26,23)(21,28,29,24)(27,30,32,31)(33,34,37,38)(35,39,42,46)(36,43,45,40)(41,48,44,47);(9,11,14,13)(10,15,12,16)(19,26)(23,25)(27,32)(30,31)(34,38)(39,46)(40,43)(47,48);(1,3,6,5)(2,7,8,4)(9,11,14,13)(10,16,12,15)(17,19,22,26)(18,23,20,25)(21,27,29,32)(24,30,28,31)(33,35,37,42)(34,39,38,46)(36,44,45,41)(40,48,43,47);(1,4,6,7)(2,3,8,5)(9,12,14,10)(11,16,13,15)(17,20,22,18)(19,23,26,25)(21,28,29,24)(27,30,32,31)(33,36)(34,40)(35,41)(37,45)(38,43)(39,47)(42,44)(46,48);(1,5,6,3)(2,7,8,4)(9,13,14,11)(10,16,12,15)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,34)(35,39)(36,40)(37,38)(41,47)(42,46)(43,45)(44,48)Polycyclic presentation:
2^9|a^2=fh,b^2=fi,c^2=fg,d^2=g,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4864353);