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 | |
1371 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,7,8)(3,4,5,6)(9,10,13,11)(12,15,16,14)(18,22)(19,25)(20,28)(30,32)(33,34,38,39)(35,40,41,36)(37,42,46,47)(43,45,48,44);(1,3,7,5)(2,4,8,6)(18,22)(20,28)(23,26)(27,31)(34,39)(35,41)(42,47)(43,48);(1,4,7,6)(2,3,8,5)(9,11,13,10)(12,15,16,14)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,35)(34,40)(36,39)(37,43)(38,41)(42,45)(44,47)(46,48);(1,5,7,3)(2,4,8,6)(9,12)(10,14)(11,15)(13,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,36,38,40)(34,41,39,35)(37,45,46,44)(42,43,47,48);(1,6,7,4)(2,3,8,5)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,37)(34,42)(35,43)(36,44)(38,46)(39,47)(40,45)(41,48)Polycyclic presentation:
2^9|a^2=fhi,b^2=fgh,c^2=hi,d^2=gh,e^2=g,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, 5638374);