Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
1907 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,9,7)(8,11,12,10)(13,14)(15,21)(16,18)(17,20)(19,23)(22,27)(24,25)(26,28)(29,30)(31,35)(32,34)(33,40)(36,39)(37,41)(38,43)(42,44);(5,6)(7,9)(8,11)(10,12)(14,16)(17,25)(19,21)(22,28)(29,31,34,39)(30,35,32,36)(33,38,41,44)(37,42,40,43);(6,7)(10,11)(13,15,18,23)(14,19,16,21)(17,22,25,28)(20,26,24,27)(29,31)(30,36)(32,35)(33,38)(34,39)(37,43)(40,42)(41,44);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,16,18,14)(15,19,23,21)(17,24,25,20)(22,26,28,27)(29,32,34,30)(31,35,39,36)(33,40,41,37)(38,42,44,43);(5,8)(6,10)(7,11)(9,12)(13,17,18,25)(14,20,16,24)(15,22,23,28)(19,26,21,27)(29,33)(30,37)(31,38)(32,40)(34,41)(35,42)(36,43)(39,44)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,c^2=gh,d^2=g,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4381359);