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:
(1,2,5,6)(3,7,8,4)(10,13)(11,15)(17,18,22,19)(20,25,27,23)(21,26,29,24)(28,30,32,31)(33,34,38,40)(35,36,42,44)(37,39,45,46)(41,43,47,48);(2,6)(4,7)(10,13)(14,16)(20,27)(23,25)(28,32)(30,31)(33,35,38,42)(34,36,40,44)(37,41,45,47)(39,43,46,48);(1,2)(3,4)(5,6)(7,8)(9,10,12,13)(11,16,15,14)(17,19,22,18)(20,25,27,23)(21,26,29,24)(28,31,32,30)(33,35)(34,36)(37,41)(38,42)(39,43)(40,44)(45,47)(46,48);(1,3)(2,4)(5,8)(6,7)(9,11,12,15)(10,14,13,16)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,36)(34,35)(37,43)(38,44)(39,41)(40,42)(45,48)(46,47);(1,4)(2,3)(5,7)(6,8)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,37,38,45)(34,39,40,46)(35,41,42,47)(36,43,44,48)Polycyclic presentation:
2^9|a^2=fgi,b^2=gh,c^2=fghi,d^2=fghi,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, 6038316);