Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1443 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(11,15)(14,16)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,30)(31,32)(33,34)(35,36)(37,41)(38,42)(39,40)(43,44)(45,47)(46,48);(2,5)(6,8)(10,13)(14,16)(18,22)(19,26)(24,28)(27,32)(33,35,38,40)(34,36,42,39)(37,45,46,43)(41,47,48,44);(9,10,12,13)(11,14,15,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,31)(29,32)(33,35)(34,39)(36,42)(37,43)(38,40)(41,47)(44,48)(45,46);(1,2)(3,6)(4,5)(7,8)(17,20,21,29)(18,24,22,28)(19,27,26,32)(23,30,25,31)(33,36,38,39)(34,40,42,35)(37,44,46,47)(41,45,48,43);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,25)(20,24)(21,22)(23,26)(27,31)(28,29)(30,32)(33,37,38,46)(34,41,42,48)(35,43,40,45)(36,44,39,47)Polycyclic presentation:
2^9|b^2=fg,c^2=gh,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6044735);