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 | |
1381 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,8,4,6)(11,13)(15,16)(17,18,22,19)(20,25,27,23)(21,26,29,24)(28,30,32,31)(33,34,38,39)(35,42,40,36)(37,41,45,46)(43,48,47,44);(1,3,5,4)(2,6,7,8)(9,10)(11,16)(12,14)(13,15)(20,27)(23,25)(28,32)(30,31)(34,39)(35,40)(37,45)(44,48);(3,4)(6,8)(9,11,14,13)(10,15,12,16)(17,19,22,18)(20,25,27,23)(21,26,29,24)(28,31,32,30)(33,35,38,40)(34,36,39,42)(37,43,45,47)(41,44,46,48);(1,4,5,3)(2,6,7,8)(9,12,14,10)(11,15,13,16)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,36)(34,40)(35,39)(37,44)(38,42)(41,47)(43,46)(45,48);(1,2)(3,8)(4,6)(5,7)(9,13,14,11)(10,15,12,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=fhi,d^2=hi,e^2=fh,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, 6038371);