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) = 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 | |
1396 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)(3,4)(5,7)(6,8)(9,10,12,15)(11,14,16,13)(18,22)(19,25)(20,28)(30,32)(33,34,38,41)(35,36,43,39)(37,40,45,47)(42,44,48,46);(1,3,5,8)(2,6,7,4)(9,11)(10,13)(12,16)(14,15)(18,22)(20,28)(23,26)(27,31)(37,45)(40,47)(42,48)(44,46);(1,3,5,8)(2,4,7,6)(9,10,12,15)(11,13,16,14)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30)(33,35,38,43)(34,39,41,36)(37,42,45,48)(40,46,47,44);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,15)(13,16)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32)(33,36,38,39)(34,35,41,43)(37,44,45,46)(40,42,47,48);(1,4)(2,3)(5,6)(7,8)(9,11)(10,14)(12,16)(13,15)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,37,38,45)(34,40,41,47)(35,42,43,48)(36,44,39,46)Polycyclic presentation:
2^9|a^2=hi,b^2=fgi,c^2=fg,d^2=g,e^2=gh,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, 4980945);