Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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:
(3,4)(7,8)(9,10,13,11)(12,15,16,14)(17,18,22,23)(19,26,24,20)(21,25,29,31)(27,30,32,28)(33,34)(35,37)(36,43)(38,40)(39,41)(42,45)(44,47)(46,48);(2,6)(7,8)(18,23)(19,24)(21,29)(28,30)(33,35,39,36)(34,37,41,43)(38,44,46,42)(40,47,48,45);(1,2,5,6)(3,7,4,8)(9,11,13,10)(12,15,16,14)(17,19,22,24)(18,20,23,26)(21,27,29,32)(25,28,31,30)(33,36,39,35)(34,37,41,43)(38,44,46,42)(40,45,48,47);(1,3,5,4)(2,7,6,8)(9,12)(10,14)(11,15)(13,16)(17,20)(18,24)(19,23)(21,30)(22,26)(25,27)(28,29)(31,32)(33,37)(34,36)(35,41)(38,45)(39,43)(40,44)(42,48)(46,47);(1,4,5,3)(2,8,6,7)(17,21)(18,25)(19,27)(20,28)(22,29)(23,31)(24,32)(26,30)(33,38,39,46)(34,40,41,48)(35,42,36,44)(37,45,43,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=fg,c^2=i,d^2=fh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5648397);