Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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 | |
1385 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)(9,10,13,12)(11,14,16,15)(17,18,21,22)(19,23,25,26)(20,24,28,29)(27,30,31,32)(33,34,38,42)(35,36,40,39)(37,45,46,41)(43,47,48,44);(2,5)(6,8)(9,11,13,16)(10,14,12,15)(18,22)(19,25)(20,28)(30,32)(35,40)(36,39)(43,48)(44,47);(9,12,13,10)(11,14,16,15)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,35)(34,39)(36,42)(37,43)(38,40)(41,47)(44,45)(46,48);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,14)(13,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,36,38,39)(34,40,42,35)(37,44,46,47)(41,48,45,43);(1,3)(2,6)(4,7)(5,8)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,37,38,46)(34,41,42,45)(35,43,40,48)(36,44,39,47)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=fgh,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5871641);