Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4462490 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1366 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)(3,7)(10,14)(12,15)(20,27)(24,29)(25,31)(30,32)(33,34,37,40)(35,41,42,38)(36,39,44,46)(43,47,48,45);(1,2,4,5)(3,6,7,8)(9,10)(11,12)(13,14)(15,16)(18,22)(23,28)(24,29)(30,32)(35,42)(36,44)(38,41)(39,46);(2,5)(6,8)(10,14)(11,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32)(33,35,37,42)(34,38,40,41)(36,43,44,48)(39,45,46,47);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,14)(13,16)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,36)(34,39)(35,43)(37,44)(38,45)(40,46)(41,47)(42,48);(1,3,4,7)(2,6,5,8)(9,12,13,15)(10,11,14,16)(17,20)(18,24)(19,25)(21,27)(22,29)(23,30)(26,31)(28,32)(33,34)(35,38)(36,39)(37,40)(41,42)(43,45)(44,46)(47,48)Polycyclic presentation:
2^9|a^2=g,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, 4370494);