Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1496 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,15)(12,14)(13,16)(17,18,22,19)(20,23,27,25)(21,24,29,26)(28,30,32,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(1,2,5,3)(4,7,8,6)(9,11)(10,15)(12,13)(14,16)(18,19)(20,27)(24,26)(28,32)(34,35)(36,43)(37,45)(46,47);(2,3)(6,7)(10,12)(11,16)(17,19,22,18)(20,23,27,25)(21,24,29,26)(28,31,32,30)(33,35,38,34)(36,41,43,39)(37,40,45,42)(44,46,48,47);(1,3,5,2)(4,7,8,6)(9,12,14,10)(11,13,16,15)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,36,38,43)(34,39,35,41)(37,44,45,48)(40,46,42,47);(1,4)(2,6)(3,7)(5,8)(9,13)(10,11)(12,16)(14,15)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)(33,37,38,45)(34,40,35,42)(36,44,43,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=fgi,b^2=fhi,c^2=f,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5145978);