Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1396 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,5,7,8)(11,12)(15,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(34,38)(39,41)(40,46)(47,48);(10,14)(15,16)(18,19)(20,27)(21,29)(30,31)(33,34,37,38)(35,41,43,39)(36,40,45,46)(42,48,44,47);(9,10,13,14)(11,15,12,16)(17,19,22,18)(20,25,27,23)(21,24,29,26)(28,30,32,31)(33,35,37,43)(34,39,38,41)(36,44,45,42)(40,48,46,47);(1,2)(3,5)(4,6)(7,8)(9,11,13,12)(10,15,14,16)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,48)(44,47)(45,46);(1,3,4,7)(2,5,6,8)(9,12,13,11)(10,16,14,15)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)(33,36,37,45)(34,40,38,46)(35,42,43,44)(39,47,41,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=fg,c^2=h,d^2=fh,e^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5296581);