Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4462490 |
| automorphism | Aut(G) order 4194304 |
| 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 | |
1374 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)(9,10)(11,15)(12,16)(13,14)(20,27)(24,29)(25,31)(30,32)(33,34,38,36)(35,39,42,43)(37,40,45,44)(41,46,47,48);(3,7)(6,8)(9,11,14,12)(10,15,13,16)(18,22)(23,28)(24,29)(30,32)(34,36)(39,43)(40,44)(46,48);(1,2,4,5)(3,6,7,8)(9,12,14,11)(10,15,13,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32)(33,35)(34,39)(36,43)(37,41)(38,42)(40,46)(44,48)(45,47);(1,3,4,7)(2,6,5,8)(9,13,14,10)(11,15,12,16)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,36,38,34)(35,39,42,43)(37,44,45,40)(41,46,47,48);(17,20)(18,24)(19,25)(21,27)(22,29)(23,30)(26,31)(28,32)(33,37,38,45)(34,40,36,44)(35,41,42,47)(39,46,43,48)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=fgh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfg,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4370483);