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) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1744 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(7,11)(9,12)(14,18)(19,21)(20,25)(26,27)(29,30)(31,37)(32,34)(33,36)(35,39)(38,44)(40,41)(42,43);(5,6,8,10)(7,9,11,12)(16,24)(20,25)(22,28)(26,27)(30,32)(33,41)(35,37)(38,42);(5,6)(7,9)(8,10)(11,12)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28)(29,31,34,39)(30,35,32,37)(33,42,41,38)(36,44,40,43);(1,2)(3,4)(13,15,17,23)(14,19,18,21)(16,22,24,28)(20,26,25,27)(29,32,34,30)(31,37,39,35)(33,40,41,36)(38,44,42,43);(5,7,8,11)(6,9,10,12)(13,16)(14,20)(15,22)(17,24)(18,25)(19,26)(21,27)(23,28)(29,33)(30,36)(31,38)(32,40)(34,41)(35,43)(37,44)(39,42)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=f,d^2=g,e^2=h,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, 4952464);