Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| 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 | |
1368 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,4)(7,8)(10,14)(15,16)(17,18,21,24)(19,22,25,29)(20,27,28,23)(26,32,31,30)(33,34)(35,36)(37,45)(38,39)(40,44)(41,47)(42,46)(43,48);(1,2,6,5)(3,7,4,8)(19,25)(22,29)(26,31)(30,32)(34,39)(35,40)(41,45)(42,48);(1,3,6,4)(2,7,5,8)(9,10,13,14)(11,15,12,16)(17,19)(18,22)(20,26)(21,25)(23,30)(24,29)(27,32)(28,31)(33,35,38,40)(34,36,39,44)(37,42,47,48)(41,43,45,46);(1,4,6,3)(2,8,5,7)(9,11,13,12)(10,15,14,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,36)(34,40)(35,39)(37,46)(38,44)(41,42)(43,47)(45,48);(1,5,6,2)(3,8,4,7)(9,12,13,11)(10,16,14,15)(17,20,21,28)(18,23,24,27)(19,26,25,31)(22,30,29,32)(33,37,38,47)(34,41,39,45)(35,42,40,48)(36,43,44,46)Polycyclic presentation:
2^9|a^2=fh,b^2=hi,c^2=fg,d^2=g,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4335752);