Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4462490 |
| 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 | |
1324 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)(9,10,13,12)(11,14,15,16)(17,18,22,21)(19,25,27,23)(20,28,29,24)(26,30,32,31)(34,38)(35,41)(36,44)(46,48);(10,12)(14,16)(20,29)(24,28)(26,32)(30,31)(33,34,37,38)(35,39,41,42)(36,40,44,45)(43,46,47,48);(9,11)(10,14)(12,16)(13,15)(17,19,22,27)(18,23,21,25)(20,26,29,32)(24,30,28,31)(33,34)(35,42)(36,40)(37,38)(39,41)(43,48)(44,45)(46,47);(1,2)(3,6)(4,5)(7,8)(9,12,13,10)(11,14,15,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,35)(34,39)(36,43)(37,41)(38,42)(40,46)(44,47)(45,48);(1,3,4,7)(2,6,5,8)(17,21,22,18)(19,25,27,23)(20,28,29,24)(26,31,32,30)(33,36,37,44)(34,40,38,45)(35,43,41,47)(39,46,42,48)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=gh,d^2=fh,e^2=i,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, 4372880);