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) = 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 | |
1296 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)(3,6)(4,7)(5,8)(12,16)(14,15)(18,22)(19,25)(24,29)(27,31)(33,34,37,40)(35,38,41,45)(36,43,44,39)(42,48,47,46);(1,3)(2,4)(5,7)(6,8)(9,10,13,11)(12,14,16,15)(20,28)(24,29)(27,31)(30,32)(35,41)(38,45)(42,47)(46,48);(1,3,5,7)(2,6,8,4)(9,11,13,10)(12,15,16,14)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30)(33,35,37,41)(34,38,40,45)(36,42,44,47)(39,46,43,48);(1,3)(2,6)(4,8)(5,7)(9,12)(10,14)(11,15)(13,16)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32)(33,36,37,44)(34,39,40,43)(35,42,41,47)(38,46,45,48);(1,4,5,6)(2,7,8,3)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48)Polycyclic presentation:
2^9|a^2=fh,b^2=fghi,c^2=fi,d^2=fgh,e^2=f,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, 5666562);