Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | |
1370 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)(19,23)(20,24)(21,22)(25,31)(26,30)(27,29)(28,32)(33,34,38,41)(35,36,43,39)(37,45,46,40)(42,47,48,44);(2,5)(6,8)(9,10,13,12)(11,15,16,14)(18,22)(23,30)(24,27)(31,32)(37,46)(40,45)(42,48)(44,47);(1,2,4,5)(3,6,7,8)(9,11,13,16)(10,14,12,15)(17,18)(19,23)(20,27)(21,22)(24,29)(25,32)(26,30)(28,31)(33,35,38,43)(34,39,41,36)(37,42,46,48)(40,47,45,44);(1,3,4,7)(2,6,5,8)(9,12,13,10)(11,15,16,14)(17,19,21,26)(18,23,22,30)(20,28,29,25)(24,32,27,31)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(17,20,21,29)(18,24,22,27)(19,25,26,28)(23,31,30,32)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,45)(43,48)Polycyclic presentation:
2^9|a^2=fhi,b^2=fghi,c^2=fgh,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,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, 4106064);