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 | |
1378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,7,8,4)(9,10,13,15)(11,14,12,16)(20,27)(24,29)(25,31)(30,32)(33,34,38,39)(35,36,41,40)(37,42,45,46)(43,44,48,47);(9,11,13,12)(10,14,15,16)(18,22)(23,28)(24,29)(30,32)(34,39)(35,41)(37,45)(44,47);(2,6)(3,8)(10,15)(14,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32)(33,35,38,41)(34,40,39,36)(37,43,45,48)(42,47,46,44);(1,3)(2,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,36,38,40)(34,41,39,35)(37,44,45,47)(42,48,46,43);(1,4,5,7)(2,3,6,8)(9,12,13,11)(10,16,15,14)(17,20)(18,24)(19,25)(21,27)(22,29)(23,30)(26,31)(28,32)(33,37,38,45)(34,42,39,46)(35,43,41,48)(36,44,40,47)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=f,d^2=h,e^2=g,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, 4859636);