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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,4)(3,7,5,8)(11,12)(15,16)(17,18,22,19)(20,25,28,23)(21,24,30,26)(27,32,29,31)(33,34)(35,38)(36,41)(37,40)(39,44)(42,45)(43,48)(46,47);(1,3,6,5)(2,7,4,8)(10,14)(15,16)(18,19)(23,25)(24,26)(31,32)(34,35)(37,45)(39,41)(43,46);(1,4,6,2)(3,7,5,8)(9,10)(11,15)(12,16)(13,14)(17,19,22,18)(20,23,28,25)(21,26,30,24)(27,31,29,32)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,48,46,47);(9,11,13,12)(10,15,14,16)(17,20,22,28)(18,23,19,25)(21,29,30,27)(24,32,26,31)(33,36,38,44)(34,39,35,41)(37,46,45,43)(40,48,42,47);(1,5,6,3)(2,7,4,8)(9,12,13,11)(10,16,14,15)(17,21,22,30)(18,24,19,26)(20,27,28,29)(23,31,25,32)(33,37)(34,40)(35,42)(36,43)(38,45)(39,47)(41,48)(44,46)Polycyclic presentation:
2^9|a^2=fghi,b^2=hi,c^2=fhi,d^2=i,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, 5853832);