Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1370 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)(3,5)(4,6)(7,8)(9,10)(11,14)(12,15)(13,16)(18,22)(20,28)(23,25)(26,32)(33,34,38,35)(36,41,43,39)(37,40,45,42)(44,47,48,46);(2,3)(4,8)(9,11,12,16)(10,13,15,14)(18,22)(20,28)(23,25)(26,32)(34,35)(36,43)(40,42)(44,48);(1,2,5,3)(4,7,8,6)(9,10,12,15)(11,14,16,13)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35,38,34)(36,39,43,41)(37,40,45,42)(44,47,48,46);(1,3,5,2)(4,7,8,6)(9,11)(10,14)(12,16)(13,15)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,36,38,43)(34,39,35,41)(37,44,45,48)(40,46,42,47);(1,4)(2,6)(3,7)(5,8)(9,10)(11,14)(12,15)(13,16)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=gh,b^2=fi,c^2=f,d^2=fhi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5214105);