Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| 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)(6,8)(10,14)(15,16)(17,18)(19,23)(20,22)(21,24)(25,27)(26,31)(28,30)(29,32)(33,34,38,36)(35,40,43,39)(37,44,46,41)(42,47,45,48);(18,20)(23,25)(24,28)(31,32)(33,35,38,43)(34,39,36,40)(37,42,46,45)(41,47,44,48);(9,10)(11,15)(12,16)(13,14)(17,19,22,27)(18,23,20,25)(21,29,30,26)(24,32,28,31)(33,35)(34,40)(36,39)(37,45)(38,43)(41,47)(42,46)(44,48);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,15,14,16)(17,20,22,18)(19,25,27,23)(21,28,30,24)(26,32,29,31)(33,36,38,34)(35,39,43,40)(37,44,46,41)(42,47,45,48);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(17,21,22,30)(18,24,20,28)(19,26,27,29)(23,31,25,32)(33,37)(34,41)(35,42)(36,44)(38,46)(39,47)(40,48)(43,45)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fh,d^2=fg,e^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4311818);