Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1369 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(6,8)(9,10,13,14)(11,12,16,15)(17,18,21,22)(19,23,27,25)(20,24,29,30)(26,31,28,32)(33,34)(35,40)(36,42)(37,39)(38,41)(43,47)(44,48)(45,46);(9,11)(10,12)(13,16)(14,15)(18,22)(23,25)(24,30)(31,32)(33,35,39,36)(34,40,37,42)(38,44,46,43)(41,48,45,47);(10,14)(11,16)(17,19)(18,23)(20,28)(21,27)(22,25)(24,32)(26,29)(30,31)(33,36,39,35)(34,40,37,42)(38,44,46,43)(41,47,45,48);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,14)(13,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,31)(29,30)(33,37,39,34)(35,42,36,40)(38,45,46,41)(43,48,44,47);(1,3,4,7)(2,6,5,8)(9,12,13,15)(10,11,14,16)(17,20,21,29)(18,24,22,30)(19,26,27,28)(23,31,25,32)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)Polycyclic presentation:
2^9|a^2=fh,b^2=gh,c^2=gh,d^2=gh,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eh,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, 5063820);