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) = C25×C4 |
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 | |
1446 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)(9,10,12,14)(11,15,16,13)(17,18,21,22)(19,23,27,25)(20,24,28,29)(26,30,32,31)(35,41)(36,44)(38,42)(39,46);(3,7)(6,8)(11,16)(13,15)(18,22)(23,25)(24,29)(30,31)(33,34,37,40)(35,42,41,38)(36,39,44,46)(43,48,47,45);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,35,37,41)(34,38,40,42)(36,43,44,47)(39,45,46,48);(1,3)(2,6)(4,7)(5,8)(9,11,12,16)(10,13,14,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,34)(35,38)(36,39)(37,40)(41,42)(43,45)(44,46)(47,48);(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,36)(34,39)(35,43)(37,44)(38,45)(40,46)(41,47)(42,48)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,d^2=fi,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, 4371460);