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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1834 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)(5,6,8,10)(7,9,11,12)(13,14,17,18)(15,19,21,26)(16,20,25,24)(22,27,23,28)(29,30)(31,35)(32,34)(33,40)(36,39)(37,41)(38,43)(42,44);(7,11)(9,12)(14,18)(15,21)(20,24)(22,23)(29,31,34,39)(30,35,32,36)(33,38,41,44)(37,42,40,43);(5,7,8,11)(6,9,10,12)(13,15)(14,19)(16,23)(17,21)(18,26)(20,28)(22,25)(24,27)(29,31)(30,36)(32,35)(33,38)(34,39)(37,43)(40,42)(41,44);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,24)(17,18)(20,25)(21,26)(22,28)(23,27)(29,32,34,30)(31,35,39,36)(33,40,41,37)(38,42,44,43);(13,16,17,25)(14,20,18,24)(15,22,21,23)(19,27,26,28)(29,33)(30,37)(31,38)(32,40)(34,41)(35,42)(36,43)(39,44)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=fgh,d^2=g,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, 3245533);