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 | |
1362 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)(11,15)(14,16)(17,18)(19,25)(20,22)(21,24)(23,27)(26,32)(28,30)(29,31)(33,34,37,38)(35,39,41,46)(36,43,45,40)(42,48,44,47);(9,10,13,12)(11,14,15,16)(18,20)(23,25)(24,28)(31,32)(34,38)(35,41)(40,43)(42,44);(9,11,13,15)(10,14,12,16)(17,19,22,27)(18,23,20,25)(21,29,30,26)(24,32,28,31)(33,35)(34,39)(36,44)(37,41)(38,46)(40,48)(42,45)(43,47);(1,2)(3,6)(4,5)(7,8)(17,20,22,18)(19,25,27,23)(21,28,30,24)(26,32,29,31)(33,34)(35,39)(36,40)(37,38)(41,46)(42,47)(43,45)(44,48);(1,3,4,7)(2,6,5,8)(9,12,13,10)(11,16,15,14)(17,21,22,30)(18,24,20,28)(19,26,27,29)(23,31,25,32)(33,36)(34,40)(35,42)(37,45)(38,43)(39,47)(41,44)(46,48)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=g,d^2=fh,e^2=fgi,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, 4314584);