Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
1304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(5,8)(11,15)(13,16)(17,18,22,19)(20,23,28,25)(21,26,29,24)(27,31,32,30)(33,34)(35,41)(36,43)(37,40)(38,39)(42,47)(44,48)(45,46);(1,2,4,6)(3,5,7,8)(18,19)(21,29)(23,25)(27,32)(34,39)(36,41)(37,45)(42,48);(1,2)(3,5)(4,6)(7,8)(9,10,12,14)(11,13,15,16)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,30,32,31)(33,35,38,43)(34,36,39,41)(37,42,45,48)(40,44,46,47);(9,11,12,15)(10,13,14,16)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,36)(34,35)(37,44)(38,41)(39,43)(40,42)(45,47)(46,48);(1,3)(2,5)(4,7)(6,8)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,37,38,45)(34,40,39,46)(35,42,43,48)(36,44,41,47)Polycyclic presentation:
2^9|a^2=fg,b^2=fgh,c^2=fgi,d^2=fi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4425547);