Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | |
1742 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)(5,6,9,8)(7,10,12,11)(13,14,17,20)(15,21,23,18)(16,24,25,19)(22,26,28,27)(29,30)(31,36)(32,40)(33,38)(34,35)(37,43)(39,42)(41,44);(5,6,9,8)(7,11,12,10)(16,25)(19,24)(22,28)(26,27)(30,35)(33,36)(37,40)(42,44);(5,7,9,12)(6,10,8,11)(13,15,17,23)(14,18,20,21)(16,22,25,28)(19,26,24,27)(29,31,34,38)(30,36,35,33)(32,41,43,39)(37,42,40,44);(1,2,3,4)(13,16,17,25)(14,19,20,24)(15,22,23,28)(18,26,21,27)(29,32,34,43)(30,37,35,40)(31,39,38,41)(33,42,36,44);(5,8,9,6)(7,11,12,10)(13,14)(15,18)(16,19)(17,20)(21,23)(22,26)(24,25)(27,28)(29,33,34,36)(30,38,35,31)(32,42,43,44)(37,41,40,39)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=fgh,d^2=i,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1621379);