Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(9,10,13,12)(11,14,16,15)(17,18,21,24)(19,22,25,29)(20,27,28,23)(26,32,31,30)(33,34,37,38)(35,39,42,46)(36,40,45,44)(41,47,43,48);(9,11,13,16)(10,14,12,15)(19,25)(22,29)(26,31)(30,32)(34,38)(39,46)(40,44)(47,48);(1,2,5,4)(3,6,7,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,35,37,42)(34,39,38,46)(36,43,45,41)(40,48,44,47);(1,3)(2,6)(4,8)(5,7)(9,12,13,10)(11,15,16,14)(17,19,21,25)(18,22,24,29)(20,26,28,31)(23,30,27,32)(33,34)(35,39)(36,44)(37,38)(40,45)(41,48)(42,46)(43,47);(1,4,5,2)(3,8,7,6)(17,20,21,28)(18,23,24,27)(19,26,25,31)(22,30,29,32)(33,36)(34,40)(35,41)(37,45)(38,44)(39,47)(42,43)(46,48)Polycyclic presentation:
2^9|a^2=fgh,b^2=h,c^2=i,d^2=gh,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3580698);