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) = C25×C4 |
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 | |
1453 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(17,18,22,19)(20,25,27,23)(21,24,30,26)(28,32,29,31)(33,34,38,36)(35,41,43,39)(37,44,46,40)(42,47,45,48);(10,13)(14,16)(18,19)(20,27)(24,26)(28,29)(33,35,38,43)(34,39,36,41)(37,42,46,45)(40,47,44,48);(1,2,5,4)(3,6,7,8)(9,10,12,13)(11,14,15,16)(17,19,22,18)(20,25,27,23)(21,24,30,26)(28,31,29,32)(33,35)(34,39)(36,41)(37,45)(38,43)(40,48)(42,46)(44,47);(1,3)(2,6)(4,8)(5,7)(17,20,22,27)(18,23,19,25)(21,29,30,28)(24,32,26,31)(33,36,38,34)(35,41,43,39)(37,40,46,44)(42,47,45,48);(1,4,5,2)(3,8,7,6)(9,11,12,15)(10,14,13,16)(17,21,22,30)(18,24,19,26)(20,28,27,29)(23,31,25,32)(33,37,38,46)(34,40,36,44)(35,42,43,45)(39,47,41,48)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fhi,d^2=f,e^2=fghi,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, 3589702);