Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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 | |
1669 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(6,9)(10,12)(13,14,18,16)(15,21,22,19)(17,24,25,20)(23,26,28,27)(29,30,34,37)(31,32,38,35)(33,41,42,36)(39,43,44,40);(1,2,3,4)(14,16)(15,22)(20,24)(23,28)(31,38)(32,35)(39,44)(40,43);(1,2)(3,4)(13,15,18,22)(14,19,16,21)(17,23,25,28)(20,26,24,27)(29,31)(30,35)(32,37)(33,39)(34,38)(36,43)(40,41)(42,44);(5,6)(7,10)(8,9)(11,12)(13,16,18,14)(15,21,22,19)(17,20,25,24)(23,26,28,27)(29,32,34,35)(30,31,37,38)(33,40,42,43)(36,39,41,44);(5,7,8,11)(6,10,9,12)(13,17,18,25)(14,20,16,24)(15,23,22,28)(19,26,21,27)(29,33,34,42)(30,36,37,41)(31,39,38,44)(32,40,35,43)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=f,d^2=g,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2580029);