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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(10,13)(11,15)(17,18,21,24)(19,25,27,22)(20,28,29,23)(26,30,32,31)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(2,5)(6,8)(10,13)(14,16)(33,34,37,40)(35,38,43,41)(36,44,45,39)(42,47,48,46);(9,10,12,13)(11,16,15,14)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,29)(30,32)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,47)(45,48);(1,2)(3,6)(4,5)(7,8)(9,11,12,15)(10,14,13,16)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,45)(47,48);(1,3)(2,6)(4,7)(5,8)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47)Polycyclic presentation:
2^9|a^2=fgh,b^2=f,c^2=h,d^2=fg,e^2=gh,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, 5885134);