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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1890 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)(10,12)(13,14)(15,19)(16,23)(17,18)(20,25)(21,28)(22,26)(24,27)(30,34)(31,37)(32,40)(43,44);(6,9)(10,12)(14,18)(19,26)(20,23)(27,28)(29,30,33,34)(31,38,37,35)(32,36,40,42)(39,44,41,43);(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,20)(17,18)(21,27)(22,26)(23,25)(24,28)(29,30)(31,35)(32,36)(33,34)(37,38)(39,43)(40,42)(41,44);(1,2)(3,4)(13,15,17,22)(14,19,18,26)(16,24,25,21)(20,28,23,27)(29,31,33,37)(30,35,34,38)(32,41,40,39)(36,44,42,43);(5,7)(6,10)(8,11)(9,12)(13,16,17,25)(14,20,18,23)(15,21,22,24)(19,27,26,28)(29,32)(30,36)(31,39)(33,40)(34,42)(35,43)(37,41)(38,44)Polycyclic presentation:
2^9|a^2=gi,b^2=g,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, 4774195);