Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
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 | |
1373 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)(12,16)(14,15)(17,18)(19,23)(20,24)(21,22)(25,30)(26,29)(27,28)(31,32)(33,34,37,38)(35,39,41,45)(36,40,44,43)(42,46,47,48);(2,5)(6,8)(9,10,13,11)(12,14,16,15)(18,22)(23,29)(24,27)(30,31)(34,38)(35,41)(40,43)(42,47);(1,2,4,5)(3,6,7,8)(9,11,13,10)(12,15,16,14)(17,18)(19,23)(20,27)(21,22)(24,28)(25,31)(26,29)(30,32)(33,34)(35,39)(36,43)(37,38)(40,44)(41,45)(42,48)(46,47);(9,12,13,16)(10,14,11,15)(17,19,21,26)(18,23,22,29)(20,25,28,32)(24,30,27,31)(33,35,37,41)(34,39,38,45)(36,42,44,47)(40,46,43,48);(1,3)(2,6)(4,7)(5,8)(17,20)(18,24)(19,25)(21,28)(22,27)(23,30)(26,32)(29,31)(33,36,37,44)(34,40,38,43)(35,42,41,47)(39,46,45,48)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=fhi,d^2=fi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4623368);