Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1389 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,5)(6,8)(9,10,13,16)(11,14,12,15)(17,18)(19,24)(20,25)(21,29)(22,23)(26,31)(27,28)(30,32)(34,35)(37,45)(39,41)(43,48);(1,2)(3,7)(4,5)(6,8)(18,23)(20,24)(26,29)(28,32)(33,34,38,35)(36,41,44,39)(37,42,45,40)(43,46,48,47);(2,5)(6,7)(9,11,13,12)(10,14,16,15)(17,19)(18,24)(20,23)(21,30)(22,25)(26,28)(27,31)(29,32)(33,35,38,34)(36,41,44,39)(37,40,45,42)(43,46,48,47);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,20,22,24)(18,25,23,19)(21,28,31,32)(26,30,29,27)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48);(1,3)(2,6)(4,8)(5,7)(9,12,13,11)(10,15,16,14)(17,21,22,31)(18,26,23,29)(19,27,25,30)(20,28,24,32)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=ghi,d^2=fgh,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5362995);