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 | |
1358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,4)(3,7,5,8)(11,12)(15,16)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29)(33,34,38,39)(35,42,40,36)(37,41,45,47)(43,46,48,44);(1,3,6,5)(2,7,4,8)(10,14)(15,16)(20,28)(23,30)(26,32)(29,31)(34,39)(35,40)(37,45)(44,46);(1,4,6,2)(3,7,5,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,35,38,40)(34,36,39,42)(37,43,45,48)(41,44,47,46);(9,11,13,12)(10,15,14,16)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(33,36)(34,40)(35,39)(37,46)(38,42)(41,43)(44,45)(47,48);(1,5,6,3)(2,7,4,8)(9,12,13,11)(10,16,14,15)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)(33,37)(34,41)(35,43)(36,44)(38,45)(39,47)(40,48)(42,46)Polycyclic presentation:
2^9|a^2=fhi,b^2=fghi,c^2=ghi,d^2=fi,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, 5853998);