Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | |
1435 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(6,8)(11,15)(14,16)(17,18,21,22)(19,25,27,23)(20,24,29,28)(26,31,32,30)(33,34,38,41)(35,39,43,36)(37,40,46,48)(42,47,45,44);(10,13)(14,16)(18,22)(23,25)(24,28)(30,31)(33,35,38,43)(34,36,41,39)(37,45,46,42)(40,47,48,44);(9,10,12,13)(11,14,15,16)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,30)(31,32)(33,35)(34,39)(36,41)(37,45)(38,43)(40,44)(42,46)(47,48);(1,2)(3,6)(4,5)(7,8)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,48,45);(1,3,4,7)(2,6,5,8)(9,11,12,15)(10,14,13,16)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)(33,37,38,46)(34,40,41,48)(35,42,43,45)(36,44,39,47)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=fh,d^2=f,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5468466);