Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | |
1306 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)(10,12)(11,15)(17,18,21,22)(19,23,25,29)(20,27,28,24)(26,32,31,30)(34,35)(39,41)(40,42)(46,47);(10,12)(14,16)(18,22)(19,25)(20,28)(30,32)(33,34,38,35)(36,41,44,39)(37,42,45,40)(43,46,48,47);(9,10,13,12)(11,16,15,14)(17,18)(19,23)(20,27)(21,22)(24,28)(25,29)(26,32)(30,31)(33,35,38,34)(36,41,44,39)(37,40,45,42)(43,46,48,47);(1,2)(3,5)(4,6)(7,8)(9,11,13,15)(10,14,12,16)(17,19)(18,23)(20,26)(21,25)(22,29)(24,30)(27,32)(28,31)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48);(1,3,4,7)(2,5,6,8)(9,12,13,10)(11,16,15,14)(17,20,21,28)(18,24,22,27)(19,26,25,31)(23,30,29,32)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=fh,d^2=fg,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5071399);