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 | |
1304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(10,13)(11,15)(17,18,21,22)(19,23,25,29)(20,27,28,24)(26,32,31,30)(34,35)(39,41)(40,42)(46,47);(3,8)(5,7)(10,13)(14,16)(18,22)(19,25)(20,28)(30,32)(33,34,38,35)(36,41,44,39)(37,42,45,40)(43,46,48,47);(9,10)(11,16)(12,13)(14,15)(17,18)(19,23)(20,24)(21,22)(25,29)(26,30)(27,28)(31,32)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,15)(13,16)(17,19,21,25)(18,23,22,29)(20,26,28,31)(24,30,27,32)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48);(1,2)(3,5)(4,6)(7,8)(17,20,21,28)(18,24,22,27)(19,26,25,31)(23,30,29,32)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=gh,d^2=fgh,e^2=f,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, 5908791);