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:
(1,2,5,3)(4,6,8,7)(9,10,14,16)(11,12,13,15)(17,18)(19,25)(20,28)(21,27)(22,23)(24,30)(26,29)(31,32)(34,38)(35,41)(40,43)(42,47);(18,23)(21,25)(24,28)(29,31)(33,34,37,38)(35,39,41,45)(36,43,44,40)(42,48,47,46);(2,3)(6,7)(9,11,14,13)(10,15,16,12)(17,19)(18,21)(20,26)(22,27)(23,25)(24,29)(28,31)(30,32)(33,34)(35,39)(36,40)(37,38)(41,45)(42,46)(43,44)(47,48);(1,3,5,2)(4,7,8,6)(9,12,14,15)(10,11,16,13)(17,20,22,30)(18,24,23,28)(19,26,27,32)(21,29,25,31)(33,35)(34,39)(36,42)(37,41)(38,45)(40,46)(43,48)(44,47);(1,4,5,8)(2,6,3,7)(9,13,14,11)(10,15,16,12)(17,21)(18,19)(20,29)(22,25)(23,27)(24,26)(28,32)(30,31)(33,36,37,44)(34,40,38,43)(35,42,41,47)(39,46,45,48)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=fi,d^2=gh,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, 4880917);