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 | |
1610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(6,9)(10,12)(13,14,18,15)(16,21,23,19)(17,20,26,22)(24,28,25,27)(29,30,33,36)(31,37,39,34)(32,40,41,35)(38,42,44,43);(1,2,3,4)(14,15)(16,23)(20,22)(24,25)(32,41)(35,40)(38,44)(42,43);(1,2)(3,4)(13,15,18,14)(16,21,23,19)(17,20,26,22)(24,27,25,28)(29,31,33,39)(30,34,36,37)(32,38,41,44)(35,42,40,43);(5,6,8,9)(7,10,11,12)(13,16,18,23)(14,19,15,21)(17,25,26,24)(20,28,22,27)(29,32,33,41)(30,35,36,40)(31,38,39,44)(34,42,37,43);(5,7,8,11)(6,10,9,12)(13,17,18,26)(14,20,15,22)(16,24,23,25)(19,27,21,28)(29,30)(31,34)(32,35)(33,36)(37,39)(38,42)(40,41)(43,44)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=fg,d^2=i,e^2=gi,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, 3636880);