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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(9,11)(10,15)(12,13)(14,16)(18,19)(23,25)(24,26)(30,31)(33,34,37,40)(35,38,42,46)(36,43,44,39)(41,47,48,45);(10,12)(11,16)(17,19,22,18)(20,23,28,25)(21,24,29,26)(27,31,32,30)(33,34)(35,38)(36,43)(37,40)(39,44)(41,47)(42,46)(45,48);(1,2)(3,5)(4,6)(7,8)(9,12,14,10)(11,13,16,15)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,35,37,42)(34,38,40,46)(36,41,44,48)(39,45,43,47);(1,3,4,7)(2,5,6,8)(9,13)(10,11)(12,16)(14,15)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,36,37,44)(34,39,40,43)(35,41,42,48)(38,45,46,47)Polycyclic presentation:
2^9|a^2=fhi,b^2=gh,c^2=fgh,d^2=h,e^2=fgi,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, 5496248);