Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1457 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(17,18,22,19)(20,25,27,23)(21,24,30,26)(28,32,29,31)(33,34)(35,39)(36,38)(37,44)(40,43)(41,45)(42,47)(46,48);(10,14)(15,16)(18,19)(20,27)(24,26)(28,29)(33,35,38,43)(34,39,36,40)(37,42,45,48)(41,46,44,47);(1,2,5,4)(3,6,8,7)(9,10,13,14)(11,15,12,16)(17,19,22,18)(20,25,27,23)(21,24,30,26)(28,31,29,32)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,46)(45,48);(1,3,5,8)(2,6,4,7)(9,11,13,12)(10,15,14,16)(17,20)(18,23)(19,25)(21,29)(22,27)(24,32)(26,31)(28,30)(33,36,38,34)(35,39,43,40)(37,44,45,41)(42,46,48,47);(1,4,5,2)(3,7,8,6)(9,12,13,11)(10,16,14,15)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)(33,37,38,45)(34,41,36,44)(35,42,43,48)(39,46,40,47)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=fhi,d^2=ghi,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3761997);