Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
1439 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(9,10,13,12)(11,14,15,16)(17,18,21,22)(19,23,25,26)(20,24,28,30)(27,31,29,32)(33,34,37,40)(35,41,43,38)(36,44,45,39)(42,46,48,47);(11,15)(14,16)(18,22)(19,25)(20,28)(31,32)(33,34,37,40)(35,41,43,38)(36,39,45,44)(42,47,48,46);(1,2,5,4)(3,6,7,8)(9,11,13,15)(10,14,12,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,30)(29,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,45)(47,48);(1,3,5,7)(2,6,4,8)(9,10,13,12)(11,14,15,16)(17,19)(18,23)(20,29)(21,25)(22,26)(24,32)(27,28)(30,31)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,4,5,2)(3,8,7,6)(9,12,13,10)(11,16,15,14)(17,20)(18,24)(19,27)(21,28)(22,30)(23,31)(25,29)(26,32)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=fhi,d^2=ghi,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3237679);