Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1410 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)(3,4)(5,8)(6,7)(11,15)(13,16)(20,27)(24,29)(25,31)(30,32)(33,34,38,39)(35,36,41,44)(37,45,47,40)(42,46,48,43);(1,3)(2,6)(4,8)(5,7)(9,10,12,14)(11,13,15,16)(18,22)(23,28)(24,29)(30,32)(34,39)(35,41)(40,45)(42,48);(1,4)(2,7)(3,8)(5,6)(17,18,21,22)(19,23,26,28)(20,24,27,29)(25,30,31,32)(33,35)(34,36)(37,42)(38,41)(39,44)(40,43)(45,46)(47,48);(1,4)(2,3)(5,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,36)(34,35)(37,46)(38,44)(39,41)(40,48)(42,45)(43,47);(1,2)(3,6)(4,7)(5,8)(17,20)(18,24)(19,25)(21,27)(22,29)(23,30)(26,31)(28,32)(33,37,38,47)(34,40,39,45)(35,42,41,48)(36,43,44,46)Polycyclic presentation:
2^9|a^2=fg,b^2=fgi,c^2=fgh,d^2=fgh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4780451);