Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
1322 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)(3,8)(4,5)(6,7)(11,15)(13,16)(18,22)(23,28)(24,29)(30,32)(33,34,38,37)(35,41,43,39)(36,44,45,40)(42,47,46,48);(2,7)(3,5)(9,10,12,14)(11,13,15,16)(19,25)(23,28)(26,31)(30,32)(34,37)(36,45)(39,41)(42,46);(1,3,6,5)(2,8,7,4)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,35)(34,39)(36,46)(37,41)(38,43)(40,48)(42,45)(44,47);(1,4,6,8)(2,3,7,5)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,36,38,45)(34,40,37,44)(35,42,43,46)(39,47,41,48);(1,5,6,3)(2,8,7,4)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,37,38,34)(35,41,43,39)(36,44,45,40)(42,48,46,47)Polycyclic presentation:
2^9|a^2=fh,b^2=fi,c^2=h,d^2=f,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4389223);