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 | |
1310 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)(9,10)(11,15)(12,13)(14,16)(17,18,22,23)(19,26,24,20)(21,25,29,30)(27,32,31,28)(33,34,38,41)(35,39,36,43)(37,45,46,40)(42,48,44,47);(2,5)(6,8)(10,12)(14,15)(18,23)(19,24)(21,29)(28,32)(33,35,38,36)(34,39,41,43)(37,42,46,44)(40,47,45,48);(9,11)(10,14)(12,15)(13,16)(17,19,22,24)(18,20,23,26)(21,27,29,31)(25,28,30,32)(33,34)(35,39)(36,43)(37,40)(38,41)(42,47)(44,48)(45,46);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,20)(18,24)(19,23)(21,28)(22,26)(25,31)(27,30)(29,32)(33,36,38,35)(34,39,41,43)(37,44,46,42)(40,47,45,48);(1,3)(2,6)(4,7)(5,8)(9,12,13,10)(11,15,16,14)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,37,38,46)(34,40,41,45)(35,42,36,44)(39,47,43,48)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,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, 5885193);