Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| 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 | |
1374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(9,10,14,15)(11,13,12,16)(19,25)(20,27)(23,28)(24,29)(33,34,37,38)(35,41,43,39)(36,40,44,45)(42,47,48,46);(9,11,14,12)(10,13,15,16)(18,22)(23,28)(24,29)(30,32)(34,38)(36,44)(39,41)(42,48);(1,3,4,8)(2,5,6,7)(9,12,14,11)(10,13,15,16)(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)(42,46)(44,45)(47,48);(1,2)(3,7)(4,6)(5,8)(9,13)(10,12)(11,15)(14,16)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=g,b^2=fghi,c^2=fg,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4860210);