Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
1508 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(19,25)(20,27)(23,28)(24,29)(33,34,38,39)(35,41,43,36)(37,40,45,46)(42,47,48,44);(2,5)(6,8)(9,10,12,14)(11,13,16,15)(18,22)(23,28)(24,29)(30,32)(34,39)(36,41)(37,45)(42,48);(3,7)(6,8)(11,16)(13,15)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,35,38,43)(34,36,39,41)(37,42,45,48)(40,44,46,47);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,36)(34,35)(37,44)(38,41)(39,43)(40,42)(45,47)(46,48);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,37,38,45)(34,40,39,46)(35,42,43,48)(36,44,41,47)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=g,d^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4384300);