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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(9,10,13,12)(11,14,15,16)(17,18,22,19)(20,25,27,23)(21,24,30,26)(28,32,29,31)(33,34)(35,40)(36,43)(37,39)(38,45)(41,46)(42,48)(44,47);(11,15)(14,16)(18,19)(20,27)(24,26)(28,29)(33,35,39,36)(34,40,37,43)(38,42,46,44)(41,47,45,48);(1,2,5,4)(3,6,8,7)(9,11)(10,14)(12,16)(13,15)(17,19,22,18)(20,25,27,23)(21,24,30,26)(28,31,29,32)(33,36,39,35)(34,40,37,43)(38,44,46,42)(41,47,45,48);(1,3,5,8)(2,6,4,7)(9,12,13,10)(11,16,15,14)(17,20)(18,23)(19,25)(21,29)(22,27)(24,32)(26,31)(28,30)(33,37,39,34)(35,40,36,43)(38,45,46,41)(42,47,44,48);(1,4,5,2)(3,7,8,6)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)(33,38)(34,41)(35,42)(36,44)(37,45)(39,46)(40,47)(43,48)Polycyclic presentation:
2^9|a^2=fghi,b^2=g,c^2=i,d^2=hi,e^2=fgi,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, 3742298);