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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,16)(11,14,12,15)(17,18)(19,23)(20,28)(21,22)(24,30)(25,26)(27,32)(29,31)(33,34,38,39)(35,40,41,36)(37,42,46,47)(43,48,45,44);(3,7)(5,8)(9,11,13,12)(10,14,16,15)(18,22)(19,25)(24,28)(27,29)(34,39)(36,40)(42,47)(44,48);(1,2,4,6)(3,5,7,8)(9,11,13,12)(10,14,16,15)(17,19,21,25)(18,23,22,26)(20,29,30,27)(24,32,28,31)(33,35)(34,40)(36,39)(37,45)(38,41)(42,44)(43,46)(47,48);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,30)(29,31)(33,36,38,40)(34,41,39,35)(37,44,46,48)(42,45,47,43);(1,3)(2,5)(4,7)(6,8)(9,12,13,11)(10,15,16,14)(17,20)(18,24)(19,27)(21,30)(22,28)(23,31)(25,29)(26,32)(33,37)(34,42)(35,43)(36,44)(38,46)(39,47)(40,48)(41,45)Polycyclic presentation:
2^9|a^2=i,b^2=fhi,c^2=fgi,d^2=fh,e^2=fhi,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, 5116649);