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 | |
1385 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)(3,6)(4,5)(7,8)(9,10,13,11)(12,15,16,14)(19,25)(20,27)(23,28)(24,29)(33,34,37,38)(35,41,43,39)(36,40,44,46)(42,48,45,47);(1,3,5,8)(2,6,4,7)(9,10,13,11)(12,14,16,15)(18,22)(23,28)(24,29)(30,32)(34,38)(36,44)(39,41)(42,45);(2,4)(6,7)(9,11,13,10)(12,15,16,14)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,34)(35,39)(36,40)(37,38)(41,43)(42,47)(44,46)(45,48);(1,4,5,2)(3,7,8,6)(9,12,13,16)(10,14,11,15)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,35,37,43)(34,39,38,41)(36,45,44,42)(40,48,46,47);(1,3)(2,6)(4,7)(5,8)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,36,37,44)(34,40,38,46)(35,42,43,45)(39,47,41,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=fghi,d^2=fhi,e^2=h,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, 5861647);