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 | |
1366 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)(3,4,8,7)(9,10,14,12)(11,16,13,15)(18,22)(19,25)(20,28)(30,32)(34,38)(39,45)(40,43)(46,47);(18,22)(23,26)(24,29)(30,32)(33,34,37,38)(35,39,42,45)(36,43,44,40)(41,47,48,46);(2,6)(3,8)(9,11,14,13)(10,15,12,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,34)(35,39)(36,40)(37,38)(41,46)(42,45)(43,44)(47,48);(1,3,5,8)(2,7,6,4)(9,12,14,10)(11,15,13,16)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32)(33,35,37,42)(34,39,38,45)(36,41,44,48)(40,46,43,47);(1,4,5,7)(2,8,6,3)(9,13,14,11)(10,15,12,16)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,36,37,44)(34,40,38,43)(35,41,42,48)(39,46,45,47)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=fgi,d^2=g,e^2=fgh,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, 4877206);