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 | |
1386 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)(3,5)(4,6)(7,8)(9,10,12,14)(11,13,15,16)(17,18)(19,21)(20,28)(22,23)(24,27)(25,29)(26,31)(30,32)(34,38)(36,44)(39,41)(42,48);(4,8)(6,7)(18,23)(20,29)(21,24)(26,32)(33,34,37,38)(35,41,43,39)(36,45,44,40)(42,46,48,47);(2,3)(6,7)(17,19,22,27)(18,24,23,21)(20,26,29,32)(25,31,28,30)(33,34)(35,39)(36,45)(37,38)(40,44)(41,43)(42,47)(46,48);(1,3,5,2)(4,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,20)(18,25)(19,26)(21,30)(22,29)(23,28)(24,31)(27,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,4)(2,6)(3,7)(5,8)(9,11,12,15)(10,13,14,16)(17,21,22,24)(18,19,23,27)(20,30,29,31)(25,26,28,32)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=fh,d^2=fg,e^2=fgi,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, 5071879);