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 | |
1394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(5,8)(10,13)(11,15)(17,18,21,24)(19,22,25,29)(20,27,28,23)(26,32,31,30)(34,35)(37,45)(39,41)(43,48);(19,25)(22,29)(26,31)(30,32)(33,34,38,35)(36,39,44,41)(37,42,45,40)(43,47,48,46);(1,2,4,6)(3,5,7,8)(9,10,12,13)(11,16,15,14)(17,19,21,25)(18,22,24,29)(20,26,28,31)(23,30,27,32)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,3)(2,5)(4,7)(6,8)(9,11,12,15)(10,14,13,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,36,38,44)(34,39,35,41)(37,43,45,48)(40,46,42,47);(17,20,21,28)(18,23,24,27)(19,26,25,31)(22,30,29,32)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fi,d^2=g,e^2=fh,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, 4876662);