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 | |
1304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,6,4,7)(11,15)(14,16)(17,18,21,22)(19,23,25,29)(20,27,28,24)(26,32,31,30)(33,34,38,41)(35,36,43,39)(37,40,45,47)(42,44,48,46);(1,3,5,4)(2,6,8,7)(10,13)(14,16)(18,22)(19,25)(20,28)(30,32)(37,45)(40,47)(42,48)(44,46);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,27)(21,22)(24,28)(25,29)(26,32)(30,31)(33,35,38,43)(34,39,41,36)(37,42,45,48)(40,46,47,44);(1,2)(3,6)(4,7)(5,8)(17,19,21,25)(18,23,22,29)(20,26,28,31)(24,30,27,32)(33,36,38,39)(34,35,41,43)(37,44,45,46)(40,42,47,48);(1,4,5,3)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=fi,b^2=fgi,c^2=i,d^2=g,e^2=fgi,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, 5108446);