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 | |
1388 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)(5,8)(9,10,12,14)(11,13,15,16)(18,22)(23,25)(24,29)(30,31)(33,34)(35,40)(36,44)(37,42)(38,39)(41,47)(43,48)(45,46);(1,2,4,6)(3,5,7,8)(11,15)(13,16)(20,28)(24,29)(26,32)(30,31)(34,39)(35,42)(41,44)(43,45);(1,2)(3,5)(4,6)(7,8)(9,11,12,15)(10,13,14,16)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,35,38,42)(34,40,39,37)(36,45,47,43)(41,48,44,46);(9,10)(11,13)(12,14)(15,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,36)(34,41)(35,43)(37,48)(38,47)(39,44)(40,46)(42,45);(1,3)(2,5)(4,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,37,38,40)(34,42,39,35)(36,46,47,48)(41,43,44,45)Polycyclic presentation:
2^9|a^2=fi,b^2=hi,c^2=gi,d^2=g,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, 5690164);