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) = 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 | |
1366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(9,10)(11,14)(12,16)(13,15)(17,18)(19,25)(20,22)(21,28)(23,27)(24,30)(26,31)(29,32)(33,34,38,39)(35,40,42,36)(37,41,46,47)(43,48,45,44);(3,7)(6,8)(9,11,13,12)(10,14,15,16)(18,20)(23,25)(24,28)(31,32)(34,39)(35,42)(41,47)(43,45);(1,2,4,5)(3,6,7,8)(9,12,13,11)(10,14,15,16)(17,19)(18,23)(20,25)(21,29)(22,27)(24,32)(26,30)(28,31)(33,35)(34,40)(36,39)(37,45)(38,42)(41,44)(43,46)(47,48);(1,3)(2,6)(4,7)(5,8)(9,10)(11,16)(12,14)(13,15)(17,20,22,18)(19,25,27,23)(21,28,30,24)(26,32,29,31)(33,36,38,40)(34,35,39,42)(37,44,46,48)(41,43,47,45);(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)(33,37,38,46)(34,41,39,47)(35,43,42,45)(36,44,40,48)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=fgh,d^2=f,e^2=h,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, 4533544);