Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1868 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)(7,11)(10,12)(13,14,18,20)(15,21,22,16)(17,25,26,19)(23,24,28,27)(29,30)(31,37)(32,40)(33,36)(34,39)(35,42)(38,43)(41,44);(6,9)(10,12)(15,22)(16,21)(23,28)(24,27)(29,30,33,36)(31,34,39,37)(32,35,42,40)(38,43,41,44);(5,6,8,9)(7,10,11,12)(13,15)(14,16)(17,23)(18,22)(19,24)(20,21)(25,27)(26,28)(29,30)(31,34)(32,35)(33,36)(37,39)(38,43)(40,42)(41,44);(1,2)(3,4)(13,16)(14,15)(17,24)(18,21)(19,23)(20,22)(25,28)(26,27)(29,31,33,39)(30,34,36,37)(32,41,42,38)(35,44,40,43);(5,7)(6,10)(8,11)(9,12)(13,17,18,26)(14,19,20,25)(15,23,22,28)(16,24,21,27)(29,32,33,42)(30,35,36,40)(31,38,39,41)(34,43,37,44)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=fgh,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4770138);