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 | |
1708 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)(7,10,11,12)(13,14,17,18)(15,21,22,19)(16,24,25,20)(23,26,28,27)(29,30,33,37)(31,35,39,34)(32,36,41,44)(38,43,40,42);(7,11)(10,12)(14,18)(15,22)(16,25)(26,27)(29,31,33,39)(30,34,37,35)(32,40,41,38)(36,43,44,42);(5,7,9,11)(6,10,8,12)(13,14)(15,21)(16,20)(17,18)(19,22)(23,27)(24,25)(26,28)(29,31)(30,35)(32,38)(33,39)(34,37)(36,43)(40,41)(42,44);(1,2)(3,4)(13,15)(14,19)(16,23)(17,22)(18,21)(20,26)(24,27)(25,28)(29,30)(31,34)(32,36)(33,37)(35,39)(38,42)(40,43)(41,44);(5,8,9,6)(7,12,11,10)(13,16,17,25)(14,20,18,24)(15,23,22,28)(19,26,21,27)(29,32)(30,36)(31,38)(33,41)(34,42)(35,43)(37,44)(39,40)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=fh,e^2=fgh,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, 1547188);