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 | |
1384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(11,15)(14,16)(17,18,21,22)(19,23,26,29)(20,24,27,30)(25,31,28,32)(33,34,38,36)(35,40,43,39)(37,41,45,44)(42,47,48,46);(2,5)(6,8)(10,13)(14,16)(18,22)(20,27)(23,29)(25,28)(33,35,38,43)(34,39,36,40)(37,42,45,48)(41,46,44,47);(1,2)(3,6)(4,5)(7,8)(9,10,12,13)(11,14,15,16)(17,18)(19,23)(20,24)(21,22)(25,31)(26,29)(27,30)(28,32)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,46)(45,48);(9,11)(10,14)(12,15)(13,16)(17,19,21,26)(18,23,22,29)(20,28,27,25)(24,32,30,31)(33,36,38,34)(35,40,43,39)(37,41,45,44)(42,46,48,47);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,20,21,27)(18,24,22,30)(19,25,26,28)(23,31,29,32)(33,37)(34,41)(35,42)(36,44)(38,45)(39,46)(40,47)(43,48)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=i,d^2=f,e^2=g,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, 5589998);