Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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 | |
1386 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)(9,10,12,14)(11,15,16,13)(17,18,21,22)(19,25,26,23)(20,28,30,24)(27,31,29,32)(33,34)(35,39)(36,42)(37,45)(38,41)(40,46)(43,48)(44,47);(2,5)(6,8)(18,22)(19,26)(24,28)(27,29)(33,35,38,36)(34,39,41,42)(37,43,46,44)(40,47,45,48);(1,2)(3,6)(4,5)(7,8)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,30)(29,32)(33,36,38,35)(34,39,41,42)(37,44,46,43)(40,47,45,48);(1,3,4,7)(2,6,5,8)(9,11,12,16)(10,13,14,15)(17,19,21,26)(18,23,22,25)(20,29,30,27)(24,32,28,31)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,47)(45,46);(17,20,21,30)(18,24,22,28)(19,27,26,29)(23,31,25,32)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)Polycyclic presentation:
2^9|a^2=fhi,b^2=fg,c^2=fg,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2561224);