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 | |
1298 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)(3,5)(4,6)(7,8)(9,10,13,11)(12,15,16,14)(20,27)(24,29)(25,31)(30,32)(33,34,38,39)(35,36,40,44)(37,41,46,47)(42,43,48,45);(1,3,4,7)(2,5,6,8)(18,22)(23,28)(24,29)(30,32)(34,39)(35,40)(41,47)(42,48);(2,6)(5,8)(9,11,13,10)(12,15,16,14)(17,18,21,22)(19,23,26,28)(20,24,27,29)(25,30,31,32)(33,35)(34,36)(37,42)(38,40)(39,44)(41,43)(45,47)(46,48);(1,2)(3,5)(4,6)(7,8)(9,12)(10,14)(11,15)(13,16)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,36)(34,40)(35,39)(37,45)(38,44)(41,42)(43,46)(47,48);(1,3)(2,5)(4,7)(6,8)(17,20,21,27)(18,24,22,29)(19,25,26,31)(23,30,28,32)(33,37)(34,41)(35,42)(36,43)(38,46)(39,47)(40,48)(44,45)Polycyclic presentation:
2^9|a^2=fi,b^2=g,c^2=fhi,d^2=fh,e^2=fh,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, 5650944);