Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | |
1298 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)(3,7)(4,6)(5,8)(9,10,13,11)(12,15,16,14)(20,27)(24,29)(25,31)(30,32)(33,34)(35,39)(36,44)(37,38)(40,45)(41,42)(43,48)(46,47);(1,2)(3,7)(4,6)(5,8)(9,11,13,10)(12,15,16,14)(18,22)(23,28)(24,29)(30,32)(34,38)(35,41)(40,44)(43,47);(1,2,4,6)(3,5,8,7)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32)(33,34)(35,42)(36,44)(37,38)(39,41)(40,45)(43,46)(47,48);(1,2)(3,5)(4,6)(7,8)(9,12)(10,14)(11,15)(13,16)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,35,37,41)(34,39,38,42)(36,43,45,47)(40,46,44,48);(1,3)(2,5)(4,8)(6,7)(17,20)(18,24)(19,25)(21,27)(22,29)(23,30)(26,31)(28,32)(33,36,37,45)(34,40,38,44)(35,43,41,47)(39,46,42,48)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=g,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,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, 4975019);