Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(6,8)(9,10,13,11)(12,15,16,14)(17,18,22,19)(20,25,28,23)(21,24,30,26)(27,32,29,31)(33,34,37,40)(35,41,42,38)(36,44,45,39)(43,46,48,47);(3,7)(6,8)(9,10,13,11)(12,14,16,15)(18,19)(23,25)(24,26)(31,32)(35,42)(36,45)(38,41)(39,44);(1,2,4,5)(3,6,7,8)(9,11,13,10)(12,15,16,14)(17,19,22,18)(20,23,28,25)(21,26,30,24)(27,31,29,32)(33,35,37,42)(34,38,40,41)(36,43,45,48)(39,46,44,47);(1,2)(3,6)(4,5)(7,8)(9,12)(10,14)(11,15)(13,16)(17,20,22,28)(18,23,19,25)(21,29,30,27)(24,32,26,31)(33,34)(35,41)(36,39)(37,40)(38,42)(43,47)(44,45)(46,48);(1,3)(2,6)(4,7)(5,8)(17,21,22,30)(18,24,19,26)(20,27,28,29)(23,31,25,32)(33,36,37,45)(34,39,40,44)(35,43,42,48)(38,46,41,47)Polycyclic presentation:
2^9|a^2=fghi,b^2=fgi,c^2=i,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5705230);