Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1718 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(7,11)(9,12)(13,14)(15,19)(16,20)(17,18)(21,26)(22,27)(23,25)(24,28)(29,30,33,36)(31,34,39,37)(32,40,41,35)(38,43,44,42);(2,4)(14,18)(15,21)(20,23)(22,24)(29,30,33,36)(31,37,39,34)(32,35,41,40)(38,43,44,42);(1,2)(3,4)(5,6,8,10)(7,9,11,12)(13,14)(15,19)(16,23)(17,18)(20,25)(21,26)(22,28)(24,27)(29,30)(31,34)(32,35)(33,36)(37,39)(38,42)(40,41)(43,44);(5,7,8,11)(6,9,10,12)(13,15,17,21)(14,19,18,26)(16,24,25,22)(20,28,23,27)(29,31,33,39)(30,34,36,37)(32,38,41,44)(35,42,40,43);(13,16,17,25)(14,20,18,23)(15,22,21,24)(19,27,26,28)(29,32)(30,35)(31,38)(33,41)(34,42)(36,40)(37,43)(39,44)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=i,d^2=fgi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3437221);