Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1754 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)(7,11)(10,12)(13,14,17,20)(15,21,23,18)(16,19,24,26)(22,27,28,25)(29,30,35,38)(31,36,32,34)(33,41,43,37)(39,42,40,44);(6,9)(10,12)(16,24)(19,26)(22,28)(25,27)(29,31,35,32)(30,34,38,36)(33,40,43,39)(37,44,41,42);(5,6)(7,10)(8,9)(11,12)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,26)(25,28)(29,32,35,31)(30,36,38,34)(33,40,43,39)(37,44,41,42);(1,2,3,4)(13,15,17,23)(14,18,20,21)(16,22,24,28)(19,25,26,27)(29,33,35,43)(30,37,38,41)(31,39,32,40)(34,42,36,44);(5,7,8,11)(6,10,9,12)(13,16,17,24)(14,19,20,26)(15,22,23,28)(18,25,21,27)(29,34,35,36)(30,32,38,31)(33,42,43,44)(37,40,41,39)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,d^2=i,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5188993);